Density enhancement methods and compositions

ABSTRACT

The present invention relates to granular composite density enhancement, and related methods and compositions. The application where the properties are valuable include but are not limited to: 1) additive manufacturing (“3D printing”) involving metallic, ceramic, cermet, polymer, plastic, or other dry or solvent-suspended powders or gels, 2) concrete materials, 3) solid propellant materials, 4) cermet materials, 5) granular armors, 6) glass-metal and glass-plastic mixtures, and 7) ceramics comprising (or manufactured using) granular composites.

FIELD OF THE INVENTION

This invention was made with government support under grant numbersDMR-0820341 and DMS-1211087 awarded by the National Science Foundation.The government has certain rights in the invention.

BACKGROUND

A granular material, or granular composite, is an accumulation ofconstituent particles, where each constituent has a pre-determinedgeometry (size and shape) that remains approximately fixed when theconstituents are placed in close proximity and pressed against oneanother, for example, by gravity. In a granular composite, theconstituents may also be suspended in a solvent or liquid or heldapproximately or exactly fixed in place by a “paste” or “glue”. Granularcomposites are ubiquitous throughout industry, research labs, and thenatural world. Common examples of granular composites in the naturalworld include dirt, sand, and gravel; common examples of man-madegranular composites include concrete, bird shot, sugar, baby powder,solid propellants, cermets, ceramics, inks, and colloids.

The physical characteristics of a granular composite depend intimatelyon the detailed multi-bodied structure that is formed through thephysical interaction of its constituent particles, and on the physicalcharacteristics of the materials that comprise the constituents. Thesecharacteristics include but are not limited to: porosity (fraction ofvoid space not filled by constituent particles), viscosity, mechanicalstrength, ductility, tensile strength, elastic modulus, bulk modulus,shear modulus, thermal conductivity, electrical conductivity, andthermal expansion coefficient. For example, a composite consisting of agiven type of material with a higher-porosity structure will generallybe less strong, thermally conductive, and electrically conductive than acomposite consisting of the same type of material but with alower-porosity structure. Or, a composite consisting of constituentsthat tend to be very rough (high coefficients of friction) andaspherical in shape will, when randomly mixed, generally form aless-dense (higher porosity) structure than a composite consisting ofconstituents of the same material but where the constituents arerelatively less rough and apsherical.

The study of granular composites and their applications has generallyfocused on measuring both the physical characteristics of a givencomposite and the geometric size, shape, and other physicalcharacteristics of its constituent particles. For example, in theconcrete industry, where crushed rock and sand are mixed with wet cement(the “paste”) in certain proportions to form concrete, a “passing curve”is often used to approximately represent the size distribution ofconstituent particles in the mixture. This “passing curve” is generatedby passing the dry mixture of sand and crashed rock (also called theaggregate) through a succession of finer and finer sieves, then plottingthe volume (or mass) fraction of aggregate that has passed through eachsieve. It is known that changing the size distribution of particles, forexample, by reducing the amount of smaller-sized aggregate (the sand, inthis case), can change the physical characteristics of the wet concrete,for example, wet concrete viscosity, and also of the dried and setconcrete, for example, concrete elastic moduli and durability. In thisway, some researchers have sought to improve concrete properties bychanging the mixing ratios of aggregates. F. de Larrard, Concreteoptimization with regard to packing density and rheology, 3rd RILEMinternational symposium on rheology of cement suspensions such as freshconcrete, France (2009). J. M. Shilstone, Jr., and J. M. Schilstone,Sr., Performance based concrete mixtures and specifications for today,Concrete International, 80-83, February (2002). F. de Larrard, Concretemixture proportioning, Routledge. N.Y. (1999). J. M. Schilstone,Concrete mixture optimization, Concrete International, 33-40, June(1990).

However, the broad problem of designing granular composites based onconstituent geometry and characteristics has not been generallytractable due to its immense complexity. The characteristics of acomposite depend not only on the detailed geometry and physicalcharacteristics of each and every component constituent, but also uponthe position, orientation, and arrangement of every particle in thecomposite. For example, a composite structure that is obtained byshaking constituents in a closed container and then pouring into anothercontainer will have a different porosity than a structure generated fromthe exact same constituents by vibrating at high frequency in acontainer. This difference can be quite large, for example, as much as50% less porosity for the vibrated preparation, and the inherentdifferences between the different porosity structures will have apronounced effect on the physical characteristics of the composite.

For example, in concrete, the mechanical strength of a concrete has beenshown to depend exponentially on the porosity of the aggregate mixture,with mixtures exhibiting less porosity being exponentially stronger.However, the viscosity, inversely related to ease of flow, also dependsexponentially on the porosity, with mixtures exhibiting less porosityflowing less well (having higher viscosity). A concrete must flow tosome extent in order to be poured at a job site, and as such moreporosity in the aggregate mixture might be required, even though moreporosity means lower strength. Another example is granular armors, wherelower porosity of the armor before molding would mean higher viscosity,making the finished armor more difficult to fabricate but also stronger.With respect to solid propellants, the thrust of a rocket dependsroughly on the square of the density (density in composites isproportional to one minus porosity) of the composite propellant.

In general, what is needed is the ability to effectively predict, designand control the structures of granular composites to provide a largedegree of control over composite physical characteristics. Inparticular, what is needed is a way to reduce the porosity of compositesin order to improve physical characteristics, and, in many cases, toreduce porosity while maintaining low enough viscosity to retain theability to be used in fabrication processes.

SUMMARY OF THE INVENTION

The present invention relates to granular composite density enhancement,and related methods and compositions. The applications where theseproperties are valuable include but are not limited to: 1) additivemanufacturing (“3D printing”) involving metallic, ceramic, cermet,polymer, plastic, or other dry or solvent-suspended powders or gels, 2)concrete materials, 3) solid propellant materials, 4) cermet materials,5) granular armors, 6) glass-metal and glass-plastic mixtures, and 7)ceramics comprising (or manufactured using) granular composites.

In one embodiment, the present invention contemplates a method offormulating so as to produce materials of low porosity. In oneembodiment, the present invention contemplates a method of making agranular composite composition, comprising: a) providing at least firstand second separate groups of at least 100 particles, each grouppossessing an average particle size V_(i) ^(avg) and a passing curve,representative of the particle volume probability density functionP_(i)(V) of the group, that exhibits one or more local maxima; and b)mixing particles from said two or more groups under conditions such thatsome combination of at least 50 particles from each group yields acombined granular composite exhibiting a combined passing curve,representative of the particle volume probability density function P(V)of the combination, wherein said combined granular composite has thefollowing features: i) at least two local maxima, the maximum occurringat the smaller volume (point) labeled V₁ and the maximum occurring atthe larger volume labeled V₂, associated with different particle groups“1” and “2” such that the ratio of V₂ ^(avg)/V₁ ^(avg) is less than orequal to 10,000, ii) at least one local minimum falling between theaforementioned two maxima such that the height of the passing curve atthe local minimum is no more than 75% of the height of the passing curveat either maxima, and iii) positive points V₁ and V_(r), with at leastone of the aforementioned local minimum falling between them andV_(r)/V₁=10,000, such that the integral of V·P(V) from V₁ to at leastone of the local minima failing between the aforementioned two maximaand meeting criterion ii) is at least 2% of the integral of V·P(V) fromV_(i) to V_(r), and such that the integral from that same local minimumto V_(r) is at least 2% of the integral of V·P(V) from V₁ to V_(r). Inone embodiment, V₂ ^(avg)/V₁ ^(avg)<=2,000 and >=25.

In one embodiment, there are more than two particle groups and themethod comprises, prior to step b), dividing particle groups intosubsets. In one embodiment, said mixing is done under conditions whichinhibit phase separation. In one embodiment, said combined compositeexhibits a porosity of less than 25%, or less than 20%, or even lessthan 15%. In one embodiment, said mixing reduces (relative) viscosity.In one embodiment, the combined composite is immersed in a solvent,paste, gel, liquid, or suspension. In one embodiment, V₂ ^(avg)/V₁^(avg) is less than or equal to (<=) 2,000 and greater than or equal to(>=) 25. In one embodiment, the method further comprises optimizationprocedures to calculate a low-porosity combination using two or more ofsaid particle groups, wherein the optimization includes obtainingporosity functions P_(i)(φ_(i,j1), φ_(i,j2) . . . φ_(i,jN)) or partialsubset porosity functions P_(j1,j2), . . . (φ_(1,j1),φ_(i,j2) . . . )for mixtures of particles groups. In one embodiment, optimizationprocedures incorporate constraints on physical characteristics of thecombined composite. In one embodiment, said mixing results in ahyperuniform structure. In one embodiment, said mixing results in anearly-hyperuniform structure. In one embodiment, said mixing is done ina container with a diameter and height at least 100 times that of thelargest particles in the largest group. In one embodiment, said mixingcomprises adding particles from the second group into the first group ofparticles. In one embodiment, said mixing results in a final percentageof particles from the group with smaller average particle volume ofapproximately 10-80%. In one embodiment, said mixing comprises addingparticles from the first group into the second group of particles. Inone embodiment, the particle size distribution for said first and secondgroups exhibit arithmetic standard deviations of less than 20%. In oneembodiment, the method further comprises c) using said granularcomposite with a final porosity of less than 25% as a powder, e.g. as apowder in laser sintering, or as a powder in laser melting, or as apowder in additive manufacturing of ceramics, or as a powder in powdermetallurgy, or as a powder for injection molding, or as a powder forproduction of granular armors, or as a powder for some other purpose. Inyet another embodiment, the method further comprises c) using thegranular composite in making concrete.

The present invention also contemplates compositions generated bymethods described herein. In one embodiment, the present inventioncontemplates a granular composite composition composed of at least 100particles, that exhibits a passing curve, representative of the particlesize (volume) probability density function P(Y), that has the followingfeatures; i) at least two local maxima occurring at volumes V_(i) andY_(j), wherein the smaller of the volumes is labeled V_(i) and thelarger of the volumes is labeled V_(j), ii) at least one local minimumoccurring at V_(i-j), such that V₁<V_(i-j)<V_(j), wherein the height ofthe passing curve at the local minimum occurring at V_(i-j) is no morethan 75% percent of the height of the passing curve at either localmaximum; iii) points V_(h-i) and V_(j-k), such thatV_(h-i)<V_(i)<V_(j)<V_(j-k), wherein the point V_(h-i) is defined as:whichever is larger of the volume of the smallest particle in thecomposite or a minimum between maxima at V_(h) and V_(i), V_(h)<V_(i),such that the maxima at V_(h) corresponds to a particle group “h”meeting all criteria i), ii), iii), iv) and v), find wherein the pointV_(j-k) is defined as: whichever is smaller of either the volume of thelargest particle in the composite or a minimum between maxima at V_(j)and V_(k), V_(j)<V_(k), such that the maxima at V_(k) corresponds to aparticle group “k” meeting all criteria i), ii), iii), iv) and v); iv)average particle volumes V_(i) ^(avg) and V_(j) ^(avg) such that V_(j)^(avg)/V_(i) ^(avg)≤10,000, of corresponding particle groups “i” and“j”, where group “i” is defined as the group containing all particleswith volumes ranging from V_(h-i) to V_(i-j), and group “j” is definedas the group containing all particles with volumes ranging from V_(i-j)to V_(j-k), and; v) points V₁ and V_(r), V₁<V_(i) ^(avg)<V_(j)^(avg)<V_(r) and V_(r)/V₁=10,000, such that the integral of V·P(V) fromV_(hi-i) to at least one of the local minima V_(i-j) falling between themaxima at V_(i) and V_(j) and meeting criterion ii) is at least 2% ofthe integral of V*P(V) from V₁ to V_(r), and such that the integral fromthat same local minimum at V_(i-j) to V_(j-k) is at least 2% of theintegral of V*P(V) from V₁ to V_(r). In one embodiment, for at least onepair of particle groups “j” and “i”, V_(j) ^(avg)>V₁ ^(avg), V_(j)^(avg)/V_(i) ^(avg)<=2,000 and ×=25. In one embodiment, for at least onepair of adjacent (by average volume) particle groups, the average numberof large-large nearest neighbors within the larger (by average volume)of the two groups is greater than or equal to one. In one embodiment,the relative volume of the smaller of at least one pair of adjacent (byaverage volume) particle groups is between 10% and 80% of the totalvolume of particles in the pair of groups. In one embodiment, themixture of particle groups in fixed amounts occupies a larger volume ofspace than the volume of space occupied by any single particle group inthat fixed amount on its own. In one embodiment, said composite exhibitsa porosity of less than 25%, and more preferably less than 20%, and evenless than 15%. In one embodiment, said granular composite is immersed ina solvent, paste, gel, liquid, or suspension. In one embodiment, thespatial phase separation of particles into similarly-sized groups doesnot occur for all groups of particles. In one embodiment, the paircorrelation function of said granular composite demonstrates anincreased probability of linear arrangements of the centers of threecontacting particles, where two particles exhibit volumes at least 25times that of the other particle or where two particles exhibit volumesat least 25 times smaller than that of the other particle.

In one embodiment, the present invention contemplates systems, layersand methods for additive manufacturing. In one embodiment, the presentinvention contemplates a system comprising a) a dispenser positionedover a target surface, said dispenser containing a granular compositehaving a porosity of 20% or less; and b) an energy source positioned totransfer energy to said composite when composite is dispensed on saidtarget surface. In one embodiment, said composite is protected againstoxidation with an inert shielding gas. In one embodiment, said compositeis a sinterable powder. In one embodiment, said composite is a fusiblepowder. In one embodiment, said composite is a meltable powder. It isnot intended that the present invention be limited to powders having aparticular melting temperature. In one embodiment, said meltable powderexhibits a melting temperature between 500 and 5000° C. It is also notintended that the present invention be limited to the nature of theparticles used to make the powders. In one embodiment, said powdercomprises metal particles. In one embodiment, said powder comprisesceramic particles. In one embodiment, said powder comprises cermetparticles. In one embodiment, said powder comprises a mixture of ceramicand metal particles. In one embodiment, said powder comprises carbideparticles. In one embodiment, said powder comprises glass particles. Thepowders can be mixtures of two or more particle types. In oneembodiment, said powder comprises a mixture of polymer and metalparticles. In one embodiment, said powder comprises a mixture of polymerand ceramic particles. In one embodiment, said powder comprises amixture of polymer and glass particles. In one embodiment, said powdercomprises a mixture of metal and glass particles. In one embodiment,said powder comprises a mixture of carbide and polymer particles. In oneembodiment, said powder comprises a mixture of carbide and metalparticles. In one embodiment, said powder comprises a mixture ofcarbide, cermet, and metal particles. In one embodiment, said powdercomprises a mixture of carbide, cermet, and polymer particles. In oneembodiment, said powder comprises a mixture of ceramic, metal, andpolymer particles. In one embodiment, said powder comprises a mixture ofmetal, glass, and polymer particles. In one embodiment, said powdercomprises a mixture of carbide, metal, and polymer particles. In oneembodiment, said powder comprises Titanium alloy particles and has aporosity of approximately 10%. In one embodiment, said powder comprisesfirst and second groups of particles, said particles of said first grouphaving an average particle volume that is at least 25 times larger thanthe average particle volume of said particles of said second group. Inone embodiment, said powder comprises first and second groups ofparticles, said particles of said first group having an average particlevolume that is between 25 and 2000 times larger than the averageparticle volume of said particles of said second group. In oneembodiment, said powder comprises a 62.8%:16.2%:16.7%:4.3% mixture (byvolume) of a first group comprising approximately 10 micron particles, asecond group comprising approximately 2 micron particles, a third groupcomprising approximately 200 nanometer particles, and a fourth groupcomprising approximately 40 nanometer particles, respectively, saidpowder having a porosity of approximately 4.4%. In one embodiment, saidenergy source is a laser. In one embodiment, said granular composite isin a solvent. In one embodiment, said granular composite is in a paste.

The present invention also contemplates layers. In one embodiment, thepresent invention contemplates a layer of a granular composite powder,said layer less than 1000 microns in thickness, said powder having aporosity of 20% or less. In one embodiment, said powder is sinterable.In one embodiment, said powder is fusible. In one embodiment, saidpowder is meltable. Again, it is not intended that the present inventionbe limited to the nature of the particles used to make the powders. Inone embodiment, said powder comprises metal particles. In oneembodiment, said powder comprises ceramic particles. In one embodiment,said powder comprises cermet particles. In one embodiment, said powdercomprises carbide particles. Again, mixtures of particles arecontemplated. In one embodiment, said power comprises a mixture ofceramic and metal particles. In one embodiment, said power comprises amixture of ceramic, metal and polymer particles. In one embodiment, saidpowder comprises Titanium alloy particles and has a porosity ofapproximately 10%. In one embodiment, said powder comprises first andsecond groups of particles, said particles of said first group having anaverage particle volume that is at least 25 times larger than theaverage particle volume of said particles of said second group. In oneembodiment, said powder comprises first and second groups of particles,said particles of said first group having an average particle volumethat is between 25 and 2000 times larger than the average particlevolume of said particles of said second group. In one embodiment, saidpowder comprises a 62.8%:16.2%:16.7%:4.3% mixture of a first groupcomprising approximately 10 micron particles, a second group comprisingapproximately 2 micron particles, a third group comprising approximately200 nanometer particles, and a fourth group comprising approximately 40nanometer particles, respectively, said powder having a porosity ofapproximately 4.4%. In one embodiment, said layer is positioned on asecond layer of a granular composite powder, said second layer less than1000 microns in thickness, said powder having a porosity of 20% or less.In one embodiment, both layers are approximately 50 microns inthickness. In one embodiment, said granular composite powder is in asolvent. In one embodiment, said granular composite powder is in apaste.

The present invention also contemplates methods for making layers. Inone embodiment, the present invention contemplates a method of producinglayers comprising the steps of: a) providing a source of a granularcomposite powder having a porosity of 20% or less; b) depositing a firstportion of said powder onto a target surface; c) depositing energy intothe powder of said first portion under conditions that said energycauses sintering, fusing or melting of the first powder portion so as tocreate a first layer: d) depositing a second portion of powder onto saidfirst layer; and e) depositing energy into the powder of said secondportion under conditions that said energy causes sintering, fusing ormelting of the second powder portion so as to create a second layerpositioned on said first layer. It is not intended that the presentinvention be limited by the energy source. In one embodiment, the energyis deposited by a laser. In one embodiment, step c) comprises focusingthe laser with at least one lens. It is not intended that the presentinvention be limited to layers of a particular thickness. However, inone embodiment, said first and second layers are less than 100 microns(or less than 50 microns) in thickness. It is not intended that thepresent invention be limited by the nature of the particles used to makethe powder. In one embodiment, said powder comprises metal particles. Inone embodiment, said powder comprises ceramic particles. In oneembodiment, said powder comprises cermet particles. In one embodiment,said powder comprises carbide particles. In one embodiment, said powdercomprises a mixture of ceramic and metal particles. In one embodiment,said powder comprises a mixture of ceramic, metal and polymer particles.In one embodiment, said powder comprises Titanium alloy particles andhas a porosity of approximately 10%. It is also not intended that thepresent invention be limited by the number of groups of particles. Inone embodiment, said powder comprises first and second groups ofparticles, said particles of said first group having an average particlevolume at least 25 times larger than the average particle volume of saidparticles of said second group. In one embodiment, said powder comprisesfirst and second groups of particles, said particles of said first grouphaving an average particle volume that is between 25 and 2000 timeslarger than the average particle volume of said particles of said secondgroup. In one embodiment, said powder comprises a 62.8%:16.2%:16.7%:4.3%mixture of a first group comprising approximately 10 micron particles, asecond group comprising approximately 2 micron particles, a third groupcomprising approximately 200 nanometer particles, and a fourth groupcomprising approximately 40 nanometer particles, respectively, saidpowder having a porosity of approximately 4.4%. In one embodiment, saidgranular composite powder is in a solvent. In one embodiment, saidgranular composite powder is suspended in a paste.

The present invention also contemplates making layers using two or moredifferent powders. In one embodiment, the present invention contemplatesa method of producing layers comprising the steps of: a) providing firstand second granular composite powders, each of said powders having aporosity of 20% or less; b) depositing said first powder onto a targetsurface; c) depositing energy into said first powder under conditionssuch that said energy causes sintering, fusing or melting of said firstpowder so as to create a first layer; d) depositing said second powderonto the first layer; and e) depositing energy into said second powdersuch that said energy causes sintering, fusing or melting of said secondpowder so as to create a second layer. Again, it is not intended thatthe present invention be limited by the energy source. In oneembodiment, said energy is deposited by a laser. Again, it is notintended that the present invention be limited to layers of a particularthickness. In one embodiment, said first and second layers are less than100 microns (or less than 50 micros) in thickness. Again, the presentinvention is not limited to particular types of particles or particlecombinations used to make the powders. In one embodiment, said firstpowder comprises metal particles. In one embodiment, said second powdercomprises ceramic particles. In one embodiment, said first powdercomprises cermet particles. In one embodiment, said second powdercomprises a mixture of ceramic and metal particles. In one embodiment,said first powder comprises Titanium alloy particles and has a porosityof approximately 10%. In one embodiment, said first powder comprisesfirst and second groups of particles, said particles of said first grouphaving an average particle volume at least 25 times larger than theaverage particle volume of said particles of said second group. In oneembodiment, said first powder comprises first and second groups ofparticles, said particles of said first group having an average particlevolume that is between 25 and 2000 times larger than the averageparticle volume of said particles of said second group.

The systems, layers and methods described above can be used for 3Dprinting. With respect to powders used in 3D printing employing anenergy source to sinter, melt, or fuse particles, lower porosity meansmore reproducible manufacture, higher thermal conductivity, and a higherefficiency of laser energy absorbed by the powder, among otheradvantages.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1A is a plot of volume probability density curves for two groups ofparticles, group 1 with a mean volume V₁ ^(avg) of 25 mm³, maximum atV₁=18 mm³, find standard deviation of (½) its mean (blue), and group 2with a mean volume V₂ ^(avg) of 1000 mm³, maximum at V₁=716 mm³, and astandard deviation of (½) its mean (green). FIG. 1B shows passing curvesfor the same two particle groups, plotted for S=1.01005, assuming thateach particle group contains the same total volume of particles. TheY-axis of FIG. 1B is labeled “Fraction of particles” to emphasize thatthe scaling of the axis is dependent on the parameter S, though if eachpoint V_(S,i) were plotted, each point would represent the volume ofparticles between the points 0.5*(V_(S,i−1)+V_(S,i)) and0.5*(V_(S,i)+V_(S,i+1)) and the units of the axis would be volume (mm³).It is of note that, due to the normalization to unity of the integral ofP(V), information about the relative number of particles in each groupis not present in FIG. 1A, or generally when comparing P(V) fordifferent groups of particles. However, a passing curve or probabilitydensity function for an already-combined composite will retain thisinformation. It is also of note that in the graphs depicted, thelocations of the maxima V₁ and V₂ are not the same, with the maxima inthe passing curves (bottom) occurring at about V₁=28.1 and V₂=1113.2.Plotting the volume probability curves as passing curves repositionsmaxima and alters height and standard deviations in such a way as toallow direct comparisons between particle groups.

FIG. 2 provides an example of a passing curve plotted with S=1.00958that includes 5 local maxima (labeled V₁ to V₅). For this distributionof particles, there are two “sufficiently sized” particle groupscorresponding to the maxima at V₂ and V₅, and represented by the volumeranges [V₁, V₃₋₄], and [V₃₋₄, V₄], respectively. These particle groupsare also adjacent, both because there are no other sufficiently sizedparticle groups between them, and because the ratio of larger to smalleraverage volume of particles in the groups is less than 10,000. Themaximum at V₁ is not associated with a particle group that issufficiently sized because the smallest minima at V₁₋₂ between themaxima at V₁ and V₂ is of a height PC_(S)(V₂₋₃) that is greater than 75%of the height PC_(S)(V₁) of the maximum at V₁. The particle groupcorresponding to the maximum at V₃ is not sufficiently sized because theintegral of P(V)V, with P(V) calculated from the passing curvePC_(S)(V_(S,i)), over the range [V₂₋₃, V₃₋₄] (alternatively, the sum ofall PC_(S)(V_(S,i)) in the same range) is less than 2% of the integralof P(V)V over [V_(i), V_(r)] (alternatively, the sum of allPC_(S)(V_(S,i)) in the same range). The maximum at V₄ is not associatedwith a particle group that is sufficiently sized because, similarly tothe maximum at V₁, the height PC_(S)(V₄₋₅) of the minimum at V₄₋₅ isgreater than 75% of the height PC_(S)(V₄) of the maximum at V₄.

FIG. 3 shows the pair correlation function g₂(r) for a binary mixture(two sizes) of spherical particles with the larger particles of diameterone and the smaller of diameter 0.45 (large to small volume ratio of11), and with relative volume fraction x_(S) of smaller particles equalto 0.267 (near the minimum in porosity for this size ratio ofparticles). Note the peaks (discontinuities) and linear behavior ing₂(r) at distances r=1.175, r=1.4, and r=1.725. These discontinuities,which can be described as a sharp maximum followed by an immediatevertical drop, represent a higher probability of linear arrangements ofthe centers of contacting particles in clusters consisting of; two smalland one large (r=1.175), one large between two smalls (r=1.4), and twolarges and one small (r=1.725). The preference for these lineararrangements of the centers of three particles is unique to DSMGstructures, but can only be easily detected for mixtures of particlegroups where each and every particle group exhibits a particularly smallsize variation about the average size, i.e., the volume distributionsP(V) for each individual particle group exhibits a small standarddeviation.

FIG. 4A shows an example passing curve for a grouping of particles thatcan be considered as two sufficiently sized groups. FIG. 4B shows theexample division of grouping of particles into two sufficiently sizedgroups

FIG. 5A shows an example of passing curves for two groupings ofparticles that should be considered as one, since no combination ofrelative volumes of particles from group one (blue, lower maximum) andgroup two (green, higher maximum) can lead to a combined passing curvewhere two sufficiently sized particle groups are present. FIG. 5B showsan example combination passing curve of groupings of particles that havebeen combined as a single group.

FIG. 6 is a schematic showing the separating of 9 particle groups j=1 .. . 9 with group average particle volumes V_(j) ^(avg) into subsetsaccording to volume ranges spanning four orders of magnitude (factor of10,000). In this case, the super-set {i} of all subsets contains 4subsets, {1,2,3,4}₁{4,5}₂, {5,6,7}₃, AND {7,8,9}₄.

FIG. 7 provides a side image (FIG. 7A) and top image (FIG. 7B) of amixture of about 17% relative volume fraction of 2 mm diameter soda-limebeads with about 83% relative volume fraction of 10 mm diametersoda-lime beads, with porosity of 21.2%. The beads have sphericityof >0.98 and coefficient of static friction <0.05, and hence simulatefrictionless spheres well.

FIG. 8 is a diagram showing an example calculation of the volumes usedin calculating the approximate mixing volume fractions φ_(j) used toreduce porosity when particle groups from different subsets are mixed.The example given is for subsets that contain one overlapping particlegroup with each adjacent subset.

DEFINITIONS

As used herein, the “TJ algorithm” refers to the method for generatingdisordered strictly jammed (mechanically stable) packings (a packing isa collection of nonoverlapping objects with specified positions) ofspheres or nonspherical objects, as described in A. B. Hopkins, F. H.Stillinger, and S. Torquato, Disordered strictly jammed binary spherepackings attain an anomalously large range of densities, Physical ReviewE 88, 022205 (2013). The TJ algorithm approaches the problem ofgenerating strictly jammed packings as an optimization problem to besolved using linear programming techniques. The objective function to beminimized in this optimization problem is chosen to be roughlyequivalent to the negative of the packing fraction, where the packingfraction is the volume of space that the objects cover.

The space employed is a deformable unit cell in d dimensions withlattice vectors M_(j)={λ₁; . . . ; λ_(d)} containing N objects underperiodic boundary conditions. Each of the N objects is composed ofdifferent sizes of spheres, such that the spheres overlap and retainfixed positions with respect to one another, but not necessarily withrespect to the other objects. Objects of any shape and size can beformed using overlapping spheres that are fixed in position relative toone another.

Starting from initial conditions of the packing of N objects at anarbitrary packing fraction, where random initial conditions at lowpacking fraction yield experimentally reproducible results, a linearprogramming problem is solved to minimize the volume of the unit cellfor limited translations and rotations of the N objects, limited shearand compression of the cell, and under linearized nonoverlap conditionsof the objects. This solution results in new coordinates andorientations for the objects, and a new unit cell with smaller volume.Using these new coordinates, orientations, and unit cell, a new linearprogramming problem is solved to minimize the unit cell under similarlimited movement and nonoverlap conditions. This process is repeateduntil no further volume reductions in the unit cell are possible. Thefinal solution of this sequential linear programming (SLP) process isguaranteed to be strictly jammed (mechanically stable).

The mathematical formulation of each linear programming problem is asfollows. In this formulation, r^(λ) _(ij)=x^(λ) _(i)−x^(λ) _(j) is thedisplacement vector for spheres with positions x^(λ) _(i), in the basisof the unit cell lattice M₈₀, between spheres i and j in the packing,Δr^(λ) _(ij)=Δx^(λ) ₁−Δx^(λ) _(j) are the change in displacement to besolved for during each SLP optimization step, and ε={ε_(k1)} is thestrain tensor associated with the unit cell, with the {ε_(k1)} alsosolved for during the SLP step. All Δx^(λ) _(i) and are bounded fromabove and below to yield a limited movement range for the spheres duringeach step that is small compared to sphere diameters.

The value minimized is the trace of the strain matrix, Tr(ε)=ε₁₁+ . . .+ε_(dd), which equivalent to the linearized change in volume of the unitcell. In addition to the upper and lower bounds on Δx^(λ) _(i) andε_(k1), and Δx^(λ) _(i) and ε_(k1) must obey the linearized nonoverlapconstraints for each pair i,j of different spheres. These constraintsare written, M_(λ)·r^(λ) _(ij)·ε·M₁·r^(λ) _(ij)+Δr^(λ) _(ij)·M^(T)_(λ)·M_(λ)·r^(λ) _(ij)≥(D² _(ij)−r^(λ) _(ij)·M^(T) _(λ)·M_(λ)·r^(λ)_(ij)), with D_(ij) the average diameter of spheres i and j. For eachnonspherical object, when spheres i and j are part of the same object,these spheres not have to obey the linearized nonoverlap constraints.Rather, their positions are fixed relative to a single reference sphere,one reference sphere per object, such that their individual displacementis determined entirely by the displacement and orientation of thereference sphere. For this purpose, two additional orientation variablesφ_(i) and θ_(i) are required for each reference sphere, such that theφ_(i) and θ_(i) for each reference sphere must be bounded from above andbelow just as are the Δx^(λ) _(i) and ε_(ki), and solved for at each SLPstep.

Friction is also incorporated into the TJ algorithm via a sphere“stickiness” probability P_(f), 0≤P_(f)≤1, and distance x_(f). When twoobjects contact one another at a certain point (i.e., fall withindistance x_(f) of one another) after an SLP step, on the next SLP step,they maintain that contact with probability P_(f) via restrictions ineach object's translation and orientation. The larger the values ofP_(f) and x_(f), the greater the friction. The value P_(f)=0 correspondsto frictionless or very low-friction objects.

In a two-phase heterogeneous medium (also called “structure”), thevariance σ_(i) ²(R) of the local volume fraction of either phase isequal to (1/ν(R))·Int(χ(r)·α(r; R)dr), where “r” is a vector ind-dimensional Euclidean space, “dr” an infinitesimal volume element inthat space, ν(R) is the volume of a sphere of radius “R” in “d”dimensions, χ(f) is the autocovariance function, α(r; R) is the scaledintersection volume, and the integral “Int” runs over the entire space.If the structure is hyperuniform, then the number variance σ_(i) ²(R)grows proportionally only as fast as (1/R)^(d+1), rather than (1/R)^(d).This is equivalent to saying that, in the limit as ∥k∥ approaches zero,the spectral density, which is the Fourier transform of theautocovariance function F[χ(k)], is equal to zero, where ∥·∥ indicatesthe Euclidean distance and F[·] the Fourier transform.

Consider a two-phase medium where the one phase consists of granularparticles of any material, type, size or composition, and the other isvoid space, a solvent, a gel, a paste, or some other type offill-material. For this medium, which is a granular composite, the localvolume fraction of phase “i” at point “z₀” for a given “R” is defined asthe fraction of space belonging to phase “i” contained within ad-dimensional sphere of radius centered at point “z₀”. The variance σ₁²(R) for phase of local volume fraction is the variance of local volumefraction over all points “z₀” in the medium. Such a variance for atwo-phase medium does not depend of which phase one considers, as isindicated in the above mathematical description of σ₁ ²(R) in terms ofthe autocovariance function χ(r) and scaled intersection volume α(r; R).

The autocovariance function χ(r) in a two-phase heterogeneous medium canbe written in terms of the two-point probability function S^(i) ₂(r) andthe volume fraction φ_(i) as χ(r)=S^(i) ₂(r)−φ_(i) ², where choosingeither phase V yields the same χ(r). The two point probability functionS^(i) ₂(r) for phase “i” is equal to the probability that the end pointsof a line segment of length ∥r∥ oriented along the direction of “r” bothfall in phase and the volume fraction φ₁ is the fraction of spacecovered by phase “i”. The scaled intersection volume α(r; R) is equal tothe union of two d-dimensional spheres of radius “R” separated bydistance “r”. For more details, see C. E. Zachary, Y. Jiao, and S.Torquato, Hyperuniform long-range correlations are a signature ofdisordered jammed hard-particle packings, Physical Review Letters 106,178001 (2011).

For a finite number of constituent particles, the limit of the spectraldensity F[χ(k)] as ∥k∥ approaches zero must be defined in more detail,as only in the limit of infinite space and infinite particles does ∥k∥reach zero. For a finite group of particles, the values of ∥k∥ chosenmust be limited such that ∥k∥>1/L, where “L” is the linear extent of thesystem size. In this case, the limit as ∥k∥ approaches zero can bedefined by fitting a curve to F[χ(k)] for the smallest few points of ∥k∥and extrapolating such that that fitted curve includes the point ∥k∥=0.A structure is defined as “nearly-hyperuniform” if the limit as ∥k∥approaches zero, in units of F[χ(k)]/<V^(t/d)>² with <V^(1/d.)> theaverage effective diameter of the constituent particles, is less thanthe value “0.01”.

The value “0.01” is chosen to reflect the degree of long-range spatialcorrelations present in certain types of systems. For example, allcrystalline and quasi-crystalline arrangements of particles arehyperuniform, as are maximally random jammed arrangements of spheres.All of these systems express long range ordering between particles: in acrystal, the position of each particle is fixed relative to the positionof its neighbors, and in a maximally random jammed arrangement ofspheres, the pair correlation function decays as −1/r^(d+1), with “r”the distance between points (see A. Donev, F. H, Stillinger, and S.Torquato, Unexpected density fluctuations in jammed disordered spherepackings, Physical Review Letters 95, 090604 (2005)). In a liquid, thepair correlation function dccays exponentially fast, and, for example,in the limit as ∥k∥ approaches zero for the hard sphere liquid, F[χ(k)]is equal to about 0.028. Generally speaking, the smaller is F[χ(k)] inthe limit as ∥k∥ approaches zero, the smaller the growth of the numbervariance σ_(i) ²(R) in “R”, and the greater the spatial correlationbetween particles at large distances.

DESCRIPTION OF THE INVENTION

A granular composite density enhancement process is described forgranular composites with constituents of all sizes, shapes, and physicalcharacteristics. The process consistently results in composites thatexhibit a combination of lower porosity and viscosity than known, ingeneral practice, to be obtainable given particles within a range ofrelative sizes, where particle size, unless otherwise stated, refers tothe volume of space occupied by a given single particle. This processalso leads to reduced phase separation of particles, where theseparation of particles into distinct size groups is a significantpractical hurdle in producing dense composites when many groups ofparticles with large size differences between groups are used ascomposite constituents. The process, which involves the tailored mixingof specially-chosen size groups of particles in targeted ratios,produces structures, and therefore a composition of matter, of a typeheretofore unidentified. These structures are distinguishable throughtheir physical properties, including porosity and viscosity, and alsothrough statistical measures including but not limited to structure paircorrelation functions, contact distributions, and volume distributionsof constituents. Specifically, certain specific features exhibited bythe aforementioned statistical measures on these structures are notexhibited by the statistical measures on those granular compositestructures that are commonly known.

This process has a number of applications due to desirable reduction inporosity, reduction in viscosity, reduction in tendency to phaseseparate, or a combination of all three factors, and due to thedesirable physical and related economic properties caused by andcorrelated to reductions in porosity, viscosity and phase separation.The desirable physical properties include but are not limited to,greater bulk modulus, elastic moduli, shear moduli, durability,hardness, flowability (case of flow), thermal conductivity, heatcapacity, electrical, conductivity, overall absorption of laser (andother photonic) energy, and overall absorption of heat (and phononicenergy), as well as reductions in interface energies with “bulk”molecular solids, thermal expansion coefficient, skin depth ofabsorption of laser (and other photonic) energy, and skin depth ofabsorption of sonic energy. The desirable economic properties includebut are not limited to, decreased cost of composite components,increased reproducibility and repeatability of processing of, and moreuniform processing of, granular composites. The applications where theseproperties are valuable include but are not limited to: 1) additivemanufacturing (“3D printing”) involving metallic, ceramic, cermet,polymer, plastic, or other dry or solvent-suspended powders or gels orslurries, 2) concrete materials, 3) solid propellant materials, 4)cermet materials, 5) granular armors, 6) glass-metal and glass-plasticmixtures, and 7) ceramics comprising (or manufactured using) granularcomposites.

In additive manufacturing, a material, often a powder, must be placed ina desired spatial form and then reacted with (usually by heating) thesolid material beneath it so that the first material changes phase andbonds with the solid, becoming solid itself. In these cases, thebenefits of lower porosity and viscosity can include but are not limitedto: more even heating and melting of the granular composite, ease ofplacement and more even placement of the composite, increased overalllaser absorption and reduced skin depth of absorption, decreased lateralscattering of energy in the composite, reduced oxidation of thecomposite, and reduced temperature gradient across the composite andconsequently across the melted and resolidified solid derived from thecomposite.

In concrete materials, lower-porosity mixtures of aggregate that stillflow at the requisite rate permit reductions in paste material requiredto fill the voids between aggregate and “glue” the aggregate together.These paste materials, generally including Portland cement, are oftenthe most expensive components of the concrete, and therefore theirreduction is highly desirable. Additionally, reductions in aggregateporosity in concrete are often correlated with exponentially increasingstrength, including bulk modulus, elastic and shear moduli, hardness andlongevity.

In solid propellant materials, which are often granular composites, areduction in porosity and associated increase in density leads toincrease in propellant thrust, which can depend on the square ofcomposite density. Increased thrust is desirable due to the increasedability to lift loads, increased speed of rocket, and other desirableadvantages.

In cermet materials, granular armors, ceramics comprising (ormanufactured using) granular composites, glass metal mixtures, and glassplastic mixtures, reduced porosity leads to greater strength includingbut not limited to greater bulk modulus, elastic and shear moduli,hardness and longevity. It generally also leads to increases indurability under thermal stress cycling. These properties are oftendesirable in these materials due to their uses as protective barriers,load-bearing structural materials, and high-temperature andstress/strain durable materials.

A. Properties of the New Composition of Matter

The study of mechanically stable multimodal mixtures of granularparticles has led to the identification of a new composition of matter.This composition of matter may exhibit properties similar to both thoseof a liquid and those of a solid, for example, in that, like a liquid,it may flow when a sufficient external force (such as gravity) deformsits equilibrium shape, but, like a solid, it may withstand small butnon-zero bulk and shear stresses without deformation. This compositionof matter may, like a powder, exhibit both solid-like and liquid-likeproperties simultaneously. This composition of matter is a granularcomposite, in the sense defined previously in this document, in that itcan appear in a powder-like form, in a slurry-like (liquid) form, or ina solid-like form when its constituents have been fixed in place by a“paste” or “glue”.

Processes for producing this composition of matter will be discussedlater. In this section, the defining features and identification of thiscomposition of matter are discussed. This composition of matter isdistinguishable through its physical properties, including porosity andviscosity, and also through statistical measures including but notlimited to structure pair correlation functions, contact distributions,and volume distributions. This section discusses certain specificfeatures exhibited by the aforementioned statistical measures on thesestructures that are not exhibited by the statistical measures ongranular composite structures that are commonly known.

This composition of matter is defined by the structure and compositionof its underlying granular constituents. The class of structures thatcomprise this composition of matter will hereafter be termed DenseSmall-size-range Multimodal Granular (DSMG) structures, as certainmembers of this class of structures are unusually dense for granularstructures considering the small range of sizes spanned by theirconstituent particles.

In the following paragraphs, it is assumed that a granular composite andDSMG structure is composed of at least approximately 100 particles.Generally. DSMG structures can be and are composed of far more than 100particles, but, due to the random nature of the mixing of constituentparticles in a granular composite, at least roughly 100 particles arenecessary for DSMG structural features to become apparent.

All DSMG structures and the mixture of their constituent particlesexhibit the following characteristics:

1. DSMG structures are composed of constituent particle sizes exhibitingvolume probability density functions P(V) (also referred to simply asvolume distributions, or particle size distributions), containing two ormore local maxima associated with adjacent sufficiently-sized particlegroups. To define particle groups that are both “adjacent” and“sufficiently sized”, it is first helpful to replot the volumedistribution of the granular composite as a specific type of passingcurve that retains the maxima and minima (though they may occur atslightly different points) present in the volume distribution P(V).Subsequently, the granular composite can be divided into groups ofparticles according to this passing curve. The replotting of a volumedistribution as a passing curve is required to view maxima and minimaoccurring at different volumes on a comparable scale.

1a. The volume distribution can be plotted as a set of points V_(S,i),where this set of points is called a passing curve PC_(S)(V_(S,i)), inthe following fashion. For a range of volumes spanning from the volumeof the smallest particle to the volume of the largest particle in thecomposite, volume intervals are selected on a geometric scale. That isto say, each interval begins at X(i)=S¹ and ends at X(i+1)=S^(i+1),where the “i” are integers i=m . . . n (with “m” and “n” either or bothpossibly negative) such that for some appropriately chosen scalar S>1,“S” to the power of “n” is greater than the volume of the largestparticle, and “S” to the power of “m” is smaller than the volume of thesmallest particle A scalar S=S₀ must be chosen to be at least smallenough such that for all S, 1<S≤S₀, the number and associated maxima andminima of sufficiently-sized of particle groups determined by S remainsconstant. This means that there will be a one-to-one correspondencebetween the local extrema (maxima and minima) in the volume distributionP(V) and the passing curve PC_(S)(V_(S,i)), where the extreme consideredare those associated with sufficiently-sized particle groups.Consequently, the corresponding local maxima and minima in both thecurves P(V) and PC_(S)(V_(S,i)) associated with sufficiently-sizedparticle groups can be spoken of interchangeably.

Provided a suitably small the integral of the volume distribution timesvolume P(V)V is taken over each interval and the result plotted at themidpoint of each interval; that is, for each i=m . . . 0 . . . n−1,Int_X(i){circumflex over ( )}X(i+1)P(V)VdV is plotted at the volumepoint V_(S,i)=(½)*(X(i)+X(i+1)). To view the relative standarddeviations (standard deviation divided by mean) of particle groups on acomparable scale, the points PC_(S)(V_(S,i)) should be plotted with thevolume axis on a logarithmic scale. The resultant passing curvePC_(S)(V_(S,i)) is similar or exactly comparable to passing curvescreated by sieving particles using standard sieves because standardsieves exhibit geometrically scaled mesh sizes. A smooth curve PC_(S)(V)can be created from the set of points PC_(S)(V_(S,i)) simply byinterpolating between points using any standard method. Both curvesPC_(S)(V) and PC_(S)(V_(S,i)) exhibit the property that the sum over alli=m . . . n of PC_(S)(V_(S,i)) is equal to the average particle volumeV^(avg), just as the integral of P(V)V from V_(S,m) to V_(S,n) is equalto the average particle volume. A replotting of a volume distributionP(V) to a passing curve PC_(S)(V_(S,i)) is depicted in FIG. 1.

1b. Once plotted, the passing curve can be simply divided into mutuallyexclusive contiguous volume ranges, and accordingly, the granularcomposite particles into size groups such that the endpoints of eachvolume range bound from below and above the size of particles within thegroup. Particles with volumes falling at the endpoints can be placed ineither group having that endpoint. The volume ranges are defined suchthat each represents a particle group of “sufficient size”, and such tothe upper and lower bounds of the range lie at a local minimum fallingat volumes between local maxima associated with particle groups ofsufficient size. An example and the definition of “sufficient size”follows. From a passing curve, described in section 1a) above, considertwo local maxima occurring at volumes V₂ and V₃, V₃>V₂, such that theaverage particle volumes V₂ ^(avg) and V₃ ^(avg) of particles in groupsassociated with their respective local maxima V₂ and V₃ obey V₃^(avg)/V₂ ^(avg)<=10,000. This latter condition is one of two necessaryfor the particle groups to be considered “adjacent”. For these localmaxima to be associated with particle groups that are sufficiently sizedrequires that: a) of all local minima occurring between the two localmaxima considered, there must be a local minimum “2-3” with sizePC_(S)(V₂₋₃) occurring at V₂₋₃, that is at most 75% of the size of thesmaller of the two local maxima sizes PC_(S)(V₂), PC_(S)(V₃) and b) oversome volume range [V₁, V_(r)], such that V₁<V₂ ^(avg)<V₃ ^(avg)<V_(r)and V_(r)/V₁=10,000, both i) the sum over all i of PC_(S)(V_(S,i)) wherethe V_(S,1) are within the range spanning a local minimum occurring atV₁₋₂ between adjacent local maxima at V₁ and V₂, and V₂₋₃, must be atleast 2% of the value of the sum over all i of PC_(S)(V_(S,i)) where theV_(S,i) are within the range [V_(b), V_(r)], and ii) the sum over all iof PC_(S)(V_(S,i)) where the V_(S,i) are within the range spanning V₂₋₃and a local minimum occurring at V₃₋₄ between adjacent local maxima atV₃ and V₄, must be at least 2% of the value of the sum over all i ofPC_(S)(V_(S,i)) where the V_(S,1) are within the range [V_(i), V_(r)].If there is no local maximum of sufficient size V_(i) smaller than V₂,then a volume V just smaller than the smallest particle size is takeninstead, and if there is no local maximum of sufficient size V₄ largerthan V₃, then a volume V just greater than the largest particle size istaken instead. If there are several local minima between a pair of localmaxima, any of the local minima can be chosen, and all must beconsidered to cheek if the criteria a) and b) can be met. It isimportant to note that criterion a), after being successively applied tothe maxima of an entire granular composite, requires that both minimaassociated with (one to each side of) a local maximum of asufficiently-sized particle group have values PC_(S)(V) that are no morethan 75% of the value of the local maximum in between them.

The second condition for the two particle groups with maxima occurringat volumes V₂ and V₃ to be “adjacent” is that there must be no otherlocal maxima in PC_(S)(V) occurring at any volume V_(bi), V₂<V_(bt)<V₃,such that both the pair of maxima occurring at volumes V₂ and V_(bt) andthe pair of maxima occurring at volumes V_(bt) and V₃ meet criteria a)and b) for being “sufficiently-sized”. Any granular composite dividedinto particle groups according to the criteria just-described, such thatall sufficiently sized groups are separated, must contain at least onepair of adjacent sufficiently-sized particle groups for the composite toform a DSMG structure. The volume probability density functions orpassing curves referenced can be determined by sieving, centrifuging,image analysis, or any other general established means. In particular,it is not necessary that the functions determined be exact, but only toof a level of accuracy commonly obtained by general established means ofmeasuring such functions. FIG. 2 contains an example of particle groupdefinition using a passing curve.

2. When mixed, spatial phase separation of particles into the particlegroups associated with local maxima in passing curves according to thecriteria described in Characteristic 1 above cannot occur for all groupsacross all spatial regions. Further, all DSMG structures must exhibitsome spatial mixing of particles from at least two adjacentsufficiently-sized groups. This means that, for composites in theirpowder form, spatial mixing will include contacts between particles indifferent groups in some regions of space within the composite where theregion height, width, and depth are at least in size on the order ofseveral lengths of the largest particles from the group withlarger-volume particles.

3. When in powder form, i.e., such that no liquid or other matrixmaterial fills the void space between constituent particles, DSMGstructures have the property that: for a structure composed of n groupsof particles G_(i) masses M_(i), with the groups defined as inCharacteristic 1 above, at least one pair of groups of particles G_(i)and G_(j) (with i and j integers ranging over all n such that i does notequal j), when mixed in masses M_(i) and M_(j), yields a volume greaterthan that of either of the volumes of the individual groups G_(i) andG_(j) on its own. As a consequence of particle group volume dependenceon preparation method, this characteristic generally requires similarmethods of preparation and measurement for the volumes of single groupsof particles and mixtures of two groups of particles. Specifically, thischaracteristic excludes from the class of DSMG structures those granularcomposites that, in powder form, exhibit the property that: every groupof particles, when mixed in the same quantities that they are mixed inthe composite, form mechanically stable structures comprising onlyparticles from the group with larger volume particles, where theparticles from the group with smaller volume particles are present onlyin the voids created by the mechanically stable structure composedsolely of particles from the group with larger particles.

4. Consider all sets of particle groups in a granular composite, withgroups defined as in characteristic 1 above, such that in each set, theaverage particle volume of the group of largest particles is no morethan 10,000 times larger than the average particle volume of the groupof smallest particles. With these divisions considered, DSMG structuresrequire:

4a. It is possible to divide at least one set such that at least onepair of adjacent particle groups in the set (when particle groups areordered by average particle volume of the group) exhibits the propertythat the volume in the “larger” of the two groups is at least 20% of thetotal volume of particles in the pair of particle groups, and such thatthe total volume of particles in the “smaller” group is at least 10% ofthe total volume of the pair of particle groups. If for every pair ofparticle groups in every set of particle groups, it is not possible tomeet these conditions, then the granular composite generally will notform a DSMG structure.

4b. For the entire composite in its powder form, considering all pairsof adjacent particle groups in all sets of particles as before, for atleast one set, the average number of larger nearest-neighbor particlesfor the particles in the larger group is greater than or equal to one. A“nearest-neighbor” with volume V_(nm) to a given “central” particle withvolume V_(c) is defined as a particle that can be put in contact withthe central particle by moving it a distance less than R_(s), whereR_(s)=((3V_(nn)/4π)^(1/3)+(3V_(c)/4π)^(1/3))/2 is the averagesphere-equivalent particle radius of the two particles. Nearestneighbors can be identified by many techniques, including “freezing” apowder structure in place using special “glues” or “pastes” meant forthat purpose and performing image analysis of cross-sectional slices ofthe resulting solid, and through careful examination, by those skilledin the art, of both the pair correlation function (obtainable viastandard scattering experiments) and volume probability densityfunctions of the composite. The method of determining nearest neighborsis not relevant to the DSMG structure, though reasonably accuratemethods are necessary to obtain reasonably accurate results.

Characteristic 1 above describes DSMG granular composites in terms ofgroups of particles of different sizes. It eliminates from the class ofDSMG structures those structures formed from unimodal size distributionsof particles (only one particle group, as defined in Characteristic 1),and reflects the requirement that a particle group have a minimum totalvolume (relative to a given range of volumes) in order for that group ofparticles to contribute to the formation of DSMG structuralcharacteristics. It also requires that the ratio of larger to smalleraverage particle size in adjacent particle groups not be too large.Characteristic 2 specifies one structural feature possessed by DSMGstructures. Characteristic 3 describes a method of determining whetheror not DSMG structural characteristics can be present in a granularcomposite composed of particle groups with some physical characteristicsspecified, via statistical measures of a structure's constituentparticles. Characteristic 4 specifies structural features exhibited byall DSMG structures, and a method of determining, using statisticalmeasures, whether or not DSMG structural features can be present in agranular composite composed of particle groups with some physicalcharacteristics specified. It eliminates from the class of DSMGstructures those granular composites that consist of only sets ofsmaller and larger particles where the larger particles are moredilutely dispersed throughout the smaller particles and only rarely veryclose to or in contact with one another.

In addition to these 4 characteristics, all DSMG structures may exhibitone or more of the following characteristics:

A. Porosity less than 25% for a granular composite formed from only twogroups of particles, with groups defined as in Characteristic 1 above.

B. Porosity less than 20% for a granular composite formed from three ormore groups of particles, with groups defined as in Characteristic 1above.

C. Hyperuniformity or near-hyperuniformity, as described in theDefinitions section, while in its powder form without solvent, paste,liquid or gel between constituent particles.

D. An increased probability of, between at least one pair of particlegroups, at least one of three roughly linear arrangements of the centersof mass of 3 particles, 2 particles from one group and 1 from the other.The three possible roughly linear arrangements are, for a group ofparticles of larger size and a group of particles of smaller size,large-large-small, large-small-large, or large-small-small. Theseclusters can be detected, for some composites, by observing the granularcomposite's pair correlation function, which is accessible viascattering experiments. An example of the detection of this feature isgiven in FIG. 3, which is a depiction of the pair correlation functionof a binary mixture of spherical particles. For a composite comprisinggroups of particles, with groups defined as in Characteristic 1 above,where the particles within individual groups vary in size and shape, thepronounced peaks in pair correlation function probability at distancesrepresenting linear clusters of 3 particles, 2 from one group and 1 fromanother, will be flattened and rounded. In many DSMG structures, thoughthis clustering will be present in the structure, the feature in thepair correlation function, displayed in FIG. 3, will not be detectabledue to this flattening and rounding.

B. Process to Produce the Composition of Matter

We here describe a process to produce granular composites with reducedporosity, reduced tendency to phase separate, and reduced viscosity,relative to those composites commonly known to be capable of beingproduced, for particles within specified ratios of largest volumeparticles to smallest volume particles. The composite structuresproduced often exhibit some or all of the structural and other physicalcharacteristics described in the previous section.

It is generally known that granular composites can be produced with avariety of different porosities. For example, considering differentarrangements of same-size spherical particles, mechanically stablecomposites can be constructed ranging from about 51% porosity to about26% porosity. In this usage, “mechanically stable” means that thecomposite will withstand some non-negligible stress or shear in anydirection without collapsing, deforming, or expanding, and thereforebehaves like a solid over some range of stresses. However, thestructures that realize the upper (51%) and lower (26%) porosity boundsmentioned are highly-ordered, meaning that exacting construction methodswould be required to create structures realizing those porosities.

Considering same-size roughly spherical particles that are mixedthoroughly by vibration or shaking and that interact via friction andgravity, porosities of about 37% to 46% are common, depending on theexact shapes of the particles and the frictional coefficients betweenthem. This “thoroughly mixed” composite is an important case toconsider, because exact construction methods are not practical or evenpossible given the extremely large numbers of particles in most granularcomposites of interest. For example, only of handful of fine sand cancontain over a billion particles, and the time required to build astructure one particle at a time from such a handful would beconsiderable.

When considering granular composites comprising particles of manydifferent sizes and shapes, thoroughly-mixed, mechanically stablecomposite structures cover a broad range of possible porosities. Forexample, experiments with particles with sizes that are roughlylog-normally distributed, have yielded porosities (roughly) of up to 60%and as low as 15%, depending on particle sizes, shapes, and frictionalinteractions, as well as the details of the composite preparation,including, for example, whether or not the composite was vibrated athigh frequency or compacted. Log-normally distributed particledistributions are important for practical applications because manymanufacturing methods and natural processes produce composites withlog-normally distributed sizes. Examples of composites exhibitingroughly log-normal distributions of particle volumes include but are notlimited to clods, dirt, sand, some types of crushed rock, and some typesof nanoparticles produced via flame or chemical methods.

Research into concrete aggregate mixtures using cement, silica ash, sandand crushed rock suggests that specifically tailored mixtures of roughlylog-normally distributed constituents can yield porosities as low as15%. Most importantly though, for the lowest porosities to be achieved,the largest particles must have a far greater volume than the smallestparticles. For example, in the research referenced, the largestparticles have over 100 trillion times the volume of the smallestparticles. F. de Larrard, Concrete mixture proportioning, Routledge,N.Y. (1999).

In the academic literature, hypothetical constructions of compositeswith nearly 0% porosity are discussed. One way that this istheoretically accomplished is by creating a continuous distribution ofparticle sizes, for example, a distribution of particle volumes with anextremely large standard deviation, such that the smallest particleshave volumes that are miniscule compared to the volumes of the largestparticles, on the order of the volume difference between an atom and aboulder. In this way, when the size distribution is chosen correctly,void spaces between larger particles can always be filled by smallerparticles, and 0% porosity can be approached (though never actuallyreached).

Another theoretical way, discussed in academic literature, tosubstantially reduce porosity is to use a discrete distribution ofparticle sizes, where the average volume of a particle in eachsuccessively smaller grouping of particles is small enough so that thesmaller particles can easily fit through the spaces in the structureformed by larger particles. We term this the “discrete large ratio”approach to reflect that the composite must consist of groups ofparticles where each group exhibits a discrete average volume that is atminimum 10,000 times larger or smaller than all other average volumes ofeach group of particles in the composite. An additional requirement ofthese groups is that the standard deviation of particle volumedistributions about the average volume not be so large that particlesizes in different groups substantially overlap. The volume ratio of10,000 reflects a minimum size disparity necessary for smaller particlesto fit into the void spaces created by a mechanically stable structureof larger particles, and for small clusters of these smaller particlesnot to form structures that interfere with the particle contacts betweenlarger particles. Such clusters are discussed, for the ease ofcomposites of frictionless binary spheres, in A. B. Hopkins, F. H.Stillinger, and S. Torquato, Disordered strictly jammed binary spherepackings attain an anomalously large range of densities, Physical ReviewE 88, 022205 (2013).

When employing the discrete large ratio approach to create compositeswith low porosity, the volume of particles from each group must bechosen in precise ratios so that each successively smaller group fillsentirely the void space between the last larger group. For example,considering four groups of low-friction spherical particles, each groupconsisting of roughly same-size particles with 100,000 less volume thana particle in next larger group, a mixture (by volume) of 64.5% largestparticles, 23.6% second-largest, 8.6% third-largest, and 3.2% smallestparticles could yield a structure with a porosity of about 2.5%. It isnotable that in this example, the volume of the largest particles is onequadrillion (one thousand trillion) times that of the smallestparticles.

It is also notable that, if vibrated or mixed after settling in agravitational environment, particle groups in composites produced usingthe discrete large ratio approach will tend to phase separate, with thesmallest group of particles on the bottom, topped by the secondsmallest, and so on finally with the largest group on top. This meansthat mixing or vibrating for increasing time periods will lead todecreasing porosity of the composite, with final porosity approaching37% for the example just-described using low-friction sphericalparticles with small particle-group standard deviations. This type ofphase separation can sometimes be avoided, most successfully when theratio of larger to smaller average particle volumes between groups ofparticles is as small as possible (for the discrete large ratioapproach, the smallest such ratio is roughly 10,000). In such cases, oneway to avoid phase separation is by fabricating each larger group ofparticles out of a material that is substantially denser than theprevious smaller group. Another way is to increase substantially thefactional interactions between particles, sometimes for sub-micronparticles via electrostatic or Van der Waals forces, though this willalso increase the overall composite porosity. Yet another approach tomitigate phase separation is to compact the combined particledistribution from above, or to employ “up-down” rather than “left-right”mixing. In general, the discrete large ratio approach described is notused in applications due to a) the cost of fabricating particles thatare very similar in size for each group, b) the difficulty infabricating particle groups where particles have volumes that are manytrillions of times larger or smaller than those particles in othergroups, and c) the difficulty in overcoming particle phase separation.

It is not yet generally known among those skilled in the art or thegeneral public that another approach is possible to substantially reducecomposite porosity, where this approach can be applied to any set ofseveral groups of particles exhibiting different size distributions,shapes, fractional interactions, or other physical properties. Thisapproach does not require individual particle volume differences thatare as large as in the discrete large ratio approach previouslydescribed. Further, the composites produced by this approach do notphase separate nearly as easily as those in the discrete large ratioapproach previously described, in part because the difference inparticle volumes between groups is not as great. This latter approach,which is described in detail herein, is termed the granular compositedensity enhancement process (GCDEP).

The GCDEP can be applied provided any two or more individual groups ofgranular composites where fractions of each group, such that eachfraction exhibits a passing curve that is representative of its group,are intended to be combined in certain ratios. Each group can consist ofparticles made from the same or different materials, and the shapes andphysical characteristics of the particles in a group can be the same ordifferent. Generally, individual groups of particles should exhibit onlyone local maximum of sufficient size in their passing curve, or a singlevolume range over which local maxima of roughly comparable size arepresent. When two or more local maxima are present in a grouping ofparticles, such that two or more sufficiently-sized particle groups canbe distinguished in the fashion described in Characteristic 1 (seeabove) for some combination of volumes of particles from thesufficiently-sized groups, then the grouping should be segregated andconsidered as two or more groups such that the segregation criteriaexplained in Characteristic 1 are met. The groups can be physicallydivided according to these criteria as well. This concept is illustratedin FIG. 4. If not physically divided, then when mixing the undividedgroup with other groups, the undivided group must be considered asseveral sufficiently sized groups where the volume ratios of particlesbetween sufficiently sized groups within the undivided group are fixed.Alternatively, in two or more groupings of particles, each of whichcannot be divided into more than one sufficiently sized particle group,if the local maxima or range of volumes over which local maxima occuroverlap such that no combination of volumes from the two groups can formtwo sufficiently sized groups, then these groups should be combined andconsidered as a single group. This concept is illustrated in FIG. 5.When considered and prepared in this way, the set of all individualparticles groups shall be said to be consistent with the “GCDEP groupcriteria”.

In this way, the GCDEP is distinct from other granular compositeporosity (and sometimes also viscosity) reduction processes thatconsider the combined volume distribution of the entire granularcomposite, rather than mixing ratios of individual groups of particles.Methods that approach mixing from the perspective of combined volumedistributions are common in many fields, including but not limited tohigh performance concrete mixture proportioning. For example, many ofthe methods discussed in academic literature, including, for example, inF. de Larrard, Concrete mixture proportioning, Routledge, N.Y. (1999)and in J. M. Schilstone, Concrete mixture optimization, ConcreteInternational, 33-40, June (1990), conclude that “gaps” placed inotherwise smooth volume distributions can increase porosity and relativeviscosity, whereas the GCDEP is based in part upon the idea thatoptimizing mixture proportioning according to gap size decreasesporosity and relative viscosity. In this and other statements, “relativeviscosity” refers to the case where a granular composite consists ofparticles suspended in a fluid or other type of matrix material, wherethe relative viscosity is the viscosity of the particles and matrix atfixed ratio of particles to matrix divided by the viscosity of thematrix alone. Other approaches, including those discussed in M. N,Mangulkar and S. S. Jamkar, Review of particle packing theories used forconcrete mix proportioning, International Journal Of Scientific &Engineering Research 4, 143-148 (2013); F. de Larrard, Concreteoptimization with regard to packing density and rheology, 3rd RILEMinternational symposium on rheology of cement suspensions such as freshconcrete, France (2009); and in F. de Larrard, Concrete mixtureproportioning, Routledge, N.Y. (1999), sometimes consider distinctgroups of particles, just as does the discrete large ratio approach.However, these approaches generally favor combinations of groups suchthat the resulting composite does not meet the maxima/minima sizecriteria discussed in Characteristic 1 (see above). When thesecomposites are not favored, the assumption is made, as it is in M.Kolonko, S. Raschdorf, and D. Wasch, A hierarchical approach to simulatethe packing density of particle mixtures on a computer, Granular Matter12, 629-643 (2010), that smaller particles will fit within the voids ofa mechanically stable structure formed by the larger particles, as isthe case with the discrete large ratio approach. However, thisassumption is inaccurate with large error in porosity estimates whenaverage particle volumes of groups are in large to small ratios of lessthan 10,000, and often leads to mixed composites that easily phaseseparate.

Some approaches considering distinct groups attempt to correct for theinaccuracy in assuming that large particles will fit within the voids ofa mechanically stable structure formed by the larger particles; however,these approaches 1) cannot accurately predict mixing ratios betweengroups that yield the smallest porosities, and 2) are not based onknowledge of the structures produced by the GCDEP and use instead, forexample, inaccurate “virtual” structures, such as discussed in F. deLarrard, Concrete optimization with regard to packing density andrheology, 3rd RILEM international symposium on rheology of cementsuspensions such as fresh concrete, France (2009); and in F, de Larrard,Concrete mixture proportioning, Routledge, N.Y. (1999). These“correcting” approaches are fundamentally distinct from than the GCDEP,and they indicate different mixture proportioning than does the GCDEP.For example, the approaches most often indicate that to producelow-porosity structures, very large gaps must be present betweenparticle groups, i.e., the ratio of large to small average particlevolumes between groups will be large. When large to small averageparticle volumes are smaller than 10,000, these approaches favorcombinations of particles such that the passing curve of the resultingcomposite do not meet the maxima/minima size criteria discussed inCharacteristic 1 (see above). Consequently, the predicted porosities andrelative viscosities of the favored composites are much higher thancould be achieved using the GCDEP. For example, the compressible packingmodel of de Larrard is known to overestimate the porosity of manygranular composites including those consisting of two or more particlegroups of roughly spherical particles, as noted in E. P. Kochler, D. W.Fowler, E. H. Foley, G. J. Rogers, S. Watanachet, and M. J. Jung,Self-consolidating concrete for precast structural applications: mixtureproportions, workability, and early-age hardened properties, Center forTransportation Research, Project 0-5134-1 (2008). This is also true oflinear packing models and extensions of these models to non-sphericalparticles, as discussed in A. B. Yu, R. P. Zou, and N. Standish,Modifying the linear packing model for predicting the porosity ofnonspherical particle mixtures, Ind. Eng. Chem. Res. 35, 3730-3741(1996).

Using the GCDEP, as the examples in this document demonstrate, contraryto the conclusions of some others, “gaps” in combined volumedistributions of granular composites lead to far lower porosities andrelative viscosities than lack of gaps, when the size of the gaps andvolume ratios of individual groups mixed to form those composites arechosen carefully. The examples also demonstrate that, using the GCDEP,porosity reductions and reductions in relative viscosity can beachieved, over a given range of particles sizes, that are greater thanthose that can be achieved by both: 1) approaches that make theassumption that smaller particles fall within the voids of amechanically stable structure formed by the larger particles, and 2)approaches that attempt to correct the error introduced by theassumption that smaller particles fall within the voids of amechanically stable structure formed by the larger particles. The GCDEPis able to accomplish these reductions via a novel approach designed toencourage the production of DSMG structures.

Particle volume probability density functions for the constituents ofany granular composite can be easily experimentally determined in manydifferent ways. One way is via standard sieving, which is useful formany types of particles generally ranging in size from the micron tocentimeter scales. For smaller particles, distributions can bedetermined using scanning electron microscopes and image analysistechniques, or by using special small-particle sieves, or by desktopcentrifuges. The sieve and centrifuge methods, along with others notdescribed here, can also be used to separate particles, shouldseparation (or combination) be required to meet the “GCDEP groupcriteria” described previously. The exact procedure used to determinevolume distributions is not relevant to the GCDEP, except insofar asincreased accuracy in the procedure can result in decreased porosity andrelative viscosity when using the process.

Hereafter, we refer to a granular composite made by combining differentamounts of particles from two different groups where the groups areconsistent with the GCDEP group criteria as a “bimodal” composite, acomposite made from three different groups consistent with the criteriaas a “trimodal” composite, four sets as “quadramodal”, etc. This conceptis most usefully applied, in terms of decreasing the porosity of thegranular composite comprising a combination of different amounts ofparticles from the individual groups, when there is little overlapbetween any two volume probability density functions (or passing curves)of individual groups of particles. However, the GCDEP can just as easilybe applied when there is significant overlap between individualdistributions, even though the result will generally be that theporosity of the combined set of particles is higher.

At its core, the GCDEP allows, provided a set comprising two or moreparticles groups conforming to the GCDEP group criteria, the combinationof these groups, in specified relative ratios of volumes, so as toachieve low porosities and relative viscosities, including a method ofcalculating, within experimental accuracies, those porosities. In thisusage as before, “relative viscosity” refers to the relative viscosityof a granular composite of particles including a matrix material fillingthe voids between constituents, where the ratio of constituents tomatrix material is fixed.

However, the GCDEP can additionally be used, in conjunction withoptimization methods, to select, from a set of three or more groups ofparticles that conforms to the GCDEP group criteria, the subset ofgroups of particles that minimizes or nearly-minimizes porosity andrelative viscosity, given all possible combinations of subsets of groupsfrom the set. This Section includes descriptions of the steps involvedin the GCDEP and examples of the actual application of these steps togroups of particles. The following Section includes a description andexamples of methods to select a subset of groups from a set of three ormore particle groups so that, using the GCDEP, porosity for the setconsidering all subsets of groups of particles will be minimized ornearly-minimized.

C. Improving the Accuracy of the GCDEP

When creating a low-porosity granular composite, before employing theGCDEP, the available particulate materials that will comprise thegranular composite can be identified. For example, in creating an ink tobe used in a 3D printing process, it is generally known of what initialmaterial the final printed part will be made. Examples include pureelements like Titanium, alloys such as stainless steel and Ti 6-4,cermets like Titanium Carbide or Gallium Nitride, and mixed-phasecomposites such as sand/aluminum. Once the details have been establishedconcerning what materials are available, a plan can be developed for howto mix them. It is important to note that the GCDEP does not prescribewhat materials be used, but only how, once chosen, in what approximateproportions they should be mixed to obtain reduced porosities andrelative viscosities.

Certain knowledge is useful for the GCDEP. This knowledge includes theapproximate volume probability density functions (or passing curves) ofthe chosen group or groups of particles, which is necessary so thatgroups of particles can be confirmed to conform to the GCDEP groupcriteria. Alternatively, as described previously, groups of particlescan be separated or combined such that the final set of groups conformsto the GCDEP group criteria. For example, in cases of a particle groupthat exhibits a volume distribution spanning a large range of particlesizes, this group can be divided into two or more different groups byremoving certain sizes of particles, such that each new particle groupis separated from other groups by gaps and such that the groups in thenew set conform to the GCDEP group criteria. It is important to notethat when these steps are taken, the more accurate the volumedistributions, the more accurate will be the GCDEP mixing ratios, andtherefore the lower the possible porosity and relative viscosities thatcan be achieved.

When using lookup tables or algorithms to complete Step 1 of the GCDEP(discussed in more detail below) rather than direct experimentation,further information above and beyond volume distributions of theparticle groups are required. This information includes details aboutparticle geometry and fractional (or other, e.g., electrostatic,Van-der-Waals) interactions between particles, as well as how theparticles will be mixed (e.g., loosely mixed under gravity or compacted,vibrated, shaken, etc.). For most known materials, this information canbe found in the relevant literature, or approximations of thisinformation can be calculated. Exact information is not necessary forthe GCDEP to be applied, but more accurate information generally leadsto reduced porosities and viscosities of the final granular compositedeveloped using the GCDEP.

For sets comprising three or more particle groups conforming to theGCDEP group criteria, before the GCDEP can be applied, the super-set ofall subsets of the set of particle groups must be provided, according tothe following definitions of subset and super-set. A subset {j}_(i)consists of, over a range [V₁, V_(r)] in volume such thatV_(r)V₁=10,000, all particle groups with group average particle volumeincluded in the range [V₁, V_(r)], where {j}_(i) requires that it is the“largest” subset “i”, i.e., such that there are no subsets with moreparticle groups than {j}_(i) that also include all of the particlegroups included in {j}_(i). By this definition, a particle group canbelong to more than one subset. The super-set {i} of all subsets {j}_(i)consists of all unique largest subsets of particle groups. This conceptis illustrated in FIG. 6. Thus, for a set composed of three of moregroups of particles, the present invention contemplates criteria fordividing the set of particle groups into subsets consistent with theGCDEP group criteria.

D. The GCDEP Steps

The GCDEP steps can be summarized for one embodiment as follows:

1. Determine the relative volumes in which the groups of particleswithin a subset will be combined so as to reduce porosity. This can beaccomplished via use of computational simulation, experimentation, orlookup tables. Examples of all three cases are provided below.2. Considering the information determined in Step 1 on subsets ofparticles (subset is defined explicitly in the previous section),calculate, according to a formula described here, the relative volumesin which the groups of particles within the entire set can be combinedto reduce porosity, including the calculation of the approximateporosity achieved.

Importantly, Step 2 is not required when particle groups form only asingle subset, and when subsets do not overlap in terms of particlegroups. However, even with multiple subsets that overlap in terms ofparticle groups, rather than use the above-described Step 2 of theGCDEP, simulations, data tables, or experimentation can be undertaken(as in Step 1) to find minimal porosity structures.

Referring now to the steps in more detail, the first required step ofthe GCDEP process is to determine, for a set of particle groupsconforming to the GCDEP group criteria, the relative volumes at whichparticle groups within each subset are mixed to minimize the porosity.This can be accomplished experimentally using a well-designedexperiment, or via lookup tables or computer simulation, assuming thatthe volume distributions, rough particle shapes, and frictionalinteraction between particles are known. The use of one computersimulation to determine the minimal porosity of two particle groupswhere each group consists of a single size of frictionless spheres isdescribed in the work of Hopkins, Stillinger, and Torquato (citedabove). The computer simulation algorithm used in Hopkins, Stillingerand Torquato can also simulate three or more particles groups, and itcan take into account non-spherical particles of any shape andfrictional interaction, though including more particle groups and highlyaspherical shapes increases computational time. The TJ algorithm isdescribed in more detail in the Definitions section.

Using the volume probability density function of the particle groups, insome cases, it is possible to approximately calculate, with littleerror, the relative group volumes at which the minimum in porosityoccurs using the discrete large ratio approach. This is the case,generally, 1) when the ratio of average particle volumes between groupsof larger to smaller particles is at least 10,000, and 2) when, formixtures of relative volumes of the groups that correspond to theminimum in porosity, the integral in the combined volume distribution ofboth groups, over the range of volume where each individual volumedistribution is greater than about 3%, includes no more than about 10%of the total volume of both groups of particles. It is important to notethat the numbers 3% and 10% in these “overlap integration guidelines”are not intended to be exact boundaries. Instead, the intent in thisapproximate description is to estimate, based on existing data, when twogroups of particles can be treated using the discrete large ratioapproach.

If the discrete large ratio approach can be employed, then the mixing oftwo groups of particles will not result in structures that clearlyexhibit some or all of the distinct characteristics of DSMG structuresdescribed earlier. However, the possibility of phase separation willneed to be considered, as phase separation occurs more often and moreeasily at greater ratios of average particle volumes. In the case of nophase separation, the total packing fraction φ_(t) (equal to one minusporosity) in the discrete large ratio approach is easily calculated as,φ_(t)=φ_(L)+(1−φ_(L))·φ_(S), with φ_(L) the packing fraction of theparticle set including only larger volume particles and φ_(S) thepacking fraction of the particle set including only smaller volumeparticles. The relative volume of particles from the smaller set x_(S)follows, as x_(S)=((1−φ₁)·φ_(S))/φ₁, and from the larger set asx_(L)=1−x_(S)=φ_(L)/φ_(t).

Experimentation can also be used to find the relative volumes ofparticle groups within a subset at which the minimal porosity willoccur, though for subsets comprising three or more particle groups,experimentation can be tedious and time consuming. Considering firstonly two groups of particles, one experimental procedure to determinerelative volumes at which groups are mixed to obtain porosity minimawill be described. It is important to note that the GCDEP does notrequire that this procedure be used; rather, any experimental procedureto determine porosity minima can be used. However, the extent of theporosity and relative viscosity reductions of the granular compositewill depend on the accuracy of the procedure employed; for this reason,it is helpful to conduct the experiment very carefully in order toensure accuracy.

In one such procedure, a container capable of very accurately measuringvolumes and an accurate scale are required. For high accuracy, thecontainer must measure in length and width (or diameter), and height atleast 100 times the average longest linear extent of particles in thelarger set of particles. Additionally, when measuring the volume ofparticles, enough particles must be placed in the container such thatthe measured height of the fill is at least 100 times the averagelongest linear extent of particles in the larger group. This is due tothe effects of the boundary walls on the composite, as smaller particleswill accumulate in large numbers in the spaces against the walls inwhich the large particles cannot be placed, and larger particles canaccumulate on top of the composite.

For example, consider the set of two particle groups, the larger withelongated particles and average volume 125 times that of the smaller,and both with standard deviations of less than 10% of their mean volume.If the ratio of volumes at which these sets are mixed to obtain minimalporosity is 25% smaller, 75% larger in a cylindrical container withdiameter and height 1000 times that of the largest linear extent of thelargest particles in the composite, then if a container is used with adiameter of only 10 times the largest linear extent of the largestparticles in the composite, the minimal porosity will be increased andthe mixing ratios found to produce this porosity will be approximately70% larger particles, 30% smaller particles. This rough approximation isbased on the assumption that near a boundary wall, within one half thelength of the average largest extent of the larger particles, thecomposite will contain by volume 50% particles from the larger set 50%particles from the smaller set. In reality, this is a very coarseapproximation, as the actual effect of the boundary, due to spatialcorrelations between particles in mechanically stable granularcomposites, has a substantial effect significantly farther from theboundary than one half the length of the average largest extent of thelarger particles. Additionally, smaller and larger particle shape,frictional interactions between particles, and frictional interactionsbetween particles and the boundaries will also play a role in boundaryeffects, as well as the method in which particles are placed in thecontainer. The best way to reduce the relative effect of the boundariesis therefore to increase the relative volume to surface ratio of thecontainer, by making the container larger and filling it with morevolume of particles.

Once the containers have been selected, the zero-porosity averagedensities of the individual particles in each set (the densitycalculated by excluding void space from volume in the densitydenominator) should be obtained. There are numerous ways to accomplishthis, particularly if the particle materials are known and theirproperties can be looked up in the literature. The method whereby thedensities are determined is not important to the GCDEP, but large errorsin density will result in larger porosities and relative viscositieswhen particle groups are mixed.

Using density data, the porosity of each individual particle groupshould be determined. This can be accomplished in many ways, includingbut not limited to by placing the container on the scale, taking accountof its mass, then filling the container with particles from a singlegroup to a height at least 100 times the average longest linear extentof the particles in that group. The difference in mass between thecontainer with particles and the container without particles, divided bythe volume of the particles in the container, divided by the averagedensity of individual particles in the group, is the packing fraction(equal to one minus the porosity). This measurement should be repeatedmany times and the results averaged to obtain higher statisticalaccuracy.

When performing volume and mass measurements using the scale andcontainer, great care must be taken to prepare each composite structurein approximately the same fashion as the previous. This fashion shouldreflect the final application in which the granular composite isexpected to be used, because composite preparation can strongly affectporosity. For example, if in the final application the granularcomposite will be compacted, then the preparation should include acompaction step. If in the final application the granular composite willbe placed in a solvent, then a solvent should be added (and appropriatecalculations to reflect the mass of this solvent taken into account).

This approach should be repeated, except now considering mixtures of thetwo particle groups. Again, care should be taken such that the mixing ofthe two particle groups reflects the mixing that will occur in the finalapplication of the composite. One can start with only particles from thelarger group, then mix in increasing mass of particles from the smallergroup, or start with particles from the smaller group and mix inincreasing mass of particles from the larger group. The volume fractionof the granular composite should be calculated after each step of addinga small amount of mass of particles to the mixture and remixing in theinitial mixing fashion. Smaller increments of mass will result in a moreaccurate determination of the ratio at which maximum density occurs. Inthis case, the volume fraction is equal to the sum of the zero-porosityvolumes of the particles from each group according to their masses inthe container, divided by the volume of the mixed composite in thecontainer. For each group of particles, the zero-porosity volumes arecalculated as the mass of the particles in the container divided by thezero-porosity average densities of the particles in that group. Thisentire experiment should be repeated many times and the results averagedin order to obtain higher statistical accuracy.

Plotting the curve of packing fractions as a function of the relativevolume of particles from either the smaller or larger particle groupyields a distinct maximum (where relative volume at each point iscalculated as the zero-porosity volume of the particles from a singlegroup according to its mass in the container divided by the sum ofzero-porosity volumes of both groups according to their masses in thecontainers). The granular composites at and around this maximum willhave structures exhibiting some or all of the characteristics of DSMGstructures, as described above. Generally, mixtures with up to about 10%less absolute relative small particle group volume fraction than themaximum, and about 30-50% greater absolute relative small particle groupvolume fraction from the maximum, will exhibit some or all of the DSMGstructural characteristics. The exact distance in units of absoluterelative small particle group volume fraction less than or greater thanthe maximum at which these features are detectable depends on the sizeand shape of particles, on volume distributions, on mixing methods, andon frictional interactions between particles.

If more than two groups of particles are to be considered, theexperimental procedure just described can be performed usingcombinations of particles from three or more groups. This will takesignificantly more time, but one trained in the art can reduce this timeby focusing on relative volume fractions, informed by the experiments onpairs of particle groups, that appear most likely to yield local maximain density. In the case of three or more groups, more than one localmaxima in density can occur; identifying the largest of these maximawill enable larger reductions in overall composite porosity and relativeviscosity.

Once the relative volumes that yield porosity minima for all subsets isdetermined, the relative volume ratios in which sets should be combinedto yield DSMG structures of low porosity and relative viscosity can bedetermined to good approximation using a simple technique. Thistechnique is the second step of the GCDEP, though it is NOT a mandatorystep, as division into subsets is not strictly necessary, this stepsimply allows a quick and fairly accurate approximation in order to savetime. When there is only a single subset of particle groups, this secondstep is entirely unnecessary, and when there is no overlap betweensubsets, i.e., no two subsets contain the same particle group, then thissecond step is mathematically trivial (though still very accurate).

In this technique, the following notation will be used: the super-set ofall subsets is {i}, where the particle group with the largest averageparticle volume belongs to set i=1, and where of all remaining subsets(excluding particle groups belonging only to set 1), the subset with theparticle group having the largest average particle volume is set i=2,and so on. The subset of all particle groups included in each set is “i”is {j}_(i), where the “j” are the particle groups such that the largestgroup is the next largest and so on. So if subset 2 includes the second,third, and fourth largest groups of particles, then {2,3,4}₂ is thenotation for this set. The fraction of space occupied by each particlegroup in a subset, as found from Step 1 of the GCDEP via mixingexperiments, simulation, lookup tables, or other means, is φ_(i,j), and,along with the void space, each subset “i” of particles obeys:1=Σ_(ji)φ_(ij)+φ_(iv), with φ_(iv) the fraction of space occupied byvoids and the summation running over all “f” in subset {j}_(i). It isimportant to note that the φ_(ij) employed in the second step of theGCDEP need not be the φ_(ij) that occur at the minimum in porosity formixing of the subset “i”. However, for the technique to be accurate, theused for each subset “i” must fall on the actual mixing curve (orsurface) as determined in Step 1 of the GCDEP. As previously discussed,DSMG structures can be formed for a range of φ_(ij) values away from theφ_(ij) values that yield the minimum in porosity, and the second step ofthe GCDEP generally accommodates this range of values.

A qualitative overview of the technique is as follows. When consideringthe combination of a pair of subsets with overlap between particlegroups, some particle groups will “interact” because they are closeenough in size, while the particles from the smallest particle groups inthe smaller subset will behave as if the particles from the largestparticle groups in the larger subset are merely the boundaries of aconfined space, i.e., these groups are “non-interacting”. With this inmind, there are volumes of free space, bounded by the largest particlegroups, that are available to be filled only by the particles from thesmallest particle groups. In the notation following, these volumes endin subscript index “1”, i.e., for subscripts {α_(i)} and subset “i”,these volumes are denoted V{α₁, α₂, . . . , α_((i−2)), α_((i−1)),1}.

The number of volume spaces in each subset associated with terms incalculations of particle group final volume fraction follows a Fibonaccisequence. The first subset calculations require only one volume, labeledV₀ (V₀ should be equal to the volume of the space to be packed, which involume fraction terms is written V₀=1). The second subset requires twovolumes, V₀₀ and V₀₁, the third subset three volumes, the fourth fivevolumes, the fifth eight volumes, and so on according to the Fibonaccisequence. In binary subscript notation, the number of subscriptsindicates the subset number “i”; the term “0” indicates the volumeavailable (and occupied) by all of the particle groups in subset “i”,and the term “1” indicates the volume available to only the smallestparticle groups in subset “i”, i.e., the groups that belong to subset“i” and not to “i−1”. Additionally, a subscript label “1” cannot beadjacent to another subscript “1”, as the volume available only to thesmallest particles in a subset cannot contain, for the next subset“i+1”, “additional” volume available only to the smallest particles inthe “i+1” subset. Following these rules in writing subscripts means thatthe number of volume terms in subset “i” corresponds to the “i+1”Fibonacci number. For example, the fourth subset contains the 5 volumesV₀₀₀₀, V₀₁₀₀, V₀₀₁₀, V₀₁₀₁, and V₀₀₀₁, and “5” is the fifth Fibonaccinumber. FIG. 8 is an illustration of the calculation of these volumesfor subsets {1,2}₁, {2,3}₂, {3,4,5}₃, {5,6,7,8}₄, and {8,9}₅.

For this technique, the porosities of single particle groups (withoutmixing) P_(j) are sometimes required. Additionally, in certain cases forcertain subsets containing three or more particle groups, informationconcerning mixtures of particle groups excluding the larger group orlarger groups from that subset can be required. This requirement isdescribed in Step 3 below. In the following described technique, thequantities φ_(j) represent the volume fraction of total space that theparticles from a given particle group will occupy in the final compositemixture. This technique tends to be slightly more accurate when eachsuccessive pair of subsets shares either no particle groups or only asingle particle group between them. Consequently, these cases will bediscussed first, and the case where any two subsets share multipleparticle groups described subsequently as a special case.

The technique is as follows:

1) Starting with the first subset i=1, set the φ_(j) for all “j” insubset 1 equal to φ_(1j).

2) When particle groups are shared between subsets, calculate allvolumes V{α_(n)} for the next subset, i.e., V₀₀ and V₀₁ for the secondsubset i=2.

a. For any volume with last two subscripts “α_((i−1)), α_(i)” ending in“0,0”, V{α₁, α₂, . . . α_((i−2)), 0.0}=V{α₁, α₂, . . .α_((i−2))0}*(φ_((i−1)j)/φ_(ij)), where “j” here refers to the particlegroup that is shared by both subsets “i” and “i−1”. For example, if i=4for adjacent subsets {4,5,6}₃ and {6,7,8,9}₄, thenV₀₀₀=V₀₀₀·(φ_(3,6)/φ_(4,6)), and V₀₁₀₀=V₀₁₀·(φ_(3,6)/φ_(4,6)). If V{α₁,α₂, . . . α_((i−2)), 0,0} is greater than V{α₁, α₂, . . . α_((i−2)), 0},then calculate V′{α₁, α₂, . . . α_((i−2)), 0,0}=V{α₁, α₂, . . .α_((i−2)), 0} and calculate V′{α₁, α₂, . . . α_((i−2)),0}=V′{α₁, α₂, . .. α_((i−2)), 0,0}/V{α₁, α₂, . . . α_((i−2)), 0,0}. Use V′{α₁, α₂, . . .α_((i−2)), 0} and V′{α₁, α₂, . . . α_((i−2)), 0,0} in place of V{α₁, α₂,. . . α_((i−2)), 0} and V{α₁, α₂, . . . α_((i−2)),0,0}, respectively, inall subsequent calculations, and recalculate any other volumes requiringV{α₁, α₂, . . . α_((i−2)), 0}, using V′{α₁, α₂, . . . α_((i−2)), 0} inits place.

b. For any volume with last two subscripts “α_((i−1)), α₁” ending in“1.0” V{α₁, α₂, . . . α_((i−2)), 1.0}=V{α₁, α₂, . . . α_((i−2)), 1},assuming there are no particle groups belonging only to subset “i−1”. Ifthere are, then the fractious of space occupied by a minimal porositymixture of all particle groups that belong to subset “i−1” but NOT tosubset “i” must be known. These fractions are written φ_(i,j), and areused to calculate V{α₁, α₂, . . . , α_((i−2)), 1,0} as V′{α₁, α₂, . . .α_((i−2)), 1,0}=V{α₁, α₂, . . . α_((i−2)),1}*(φ_((i−1)); j/φ_(ij)) where“j” refers to the particle group shared by both subsets “i” and “i−1”For example, for “i=4” and subsets {2,3,4}₂, {4,5,6}₃, and {6,7,8,9}₄,V₀₀₁₀=V₀₀₁·(φ_(3:6)/φ_(4,6)), where φ_(3:6) represents the volumefraction of particle group 6 that minimizes the porosity of a mixture ofparticle groups 5 and 6.

c. For any volume with last two subscripts “α_((i−1)), α_(i)” ending in“0.1”, V{α_(i), . . . , α_((i−2)), 0.1}=V{α₁, . . . ,α_((i−1))0}*(1−Σφ_((i−1)j))−V{α₁, . . . , α_((i−2)), 0,0}, where thesummation over φ_((i−1)j) includes all such that particle group “j”belongs to subset “i−1” but not to subset “i”. For example, if i=4 forsubsets {2,3,4}₂, {4,5,6}₃, and {6,7,8,9}₄, thenV₀₁₀₁=V₀₁₀*(1−φ_(3,4)−φ_(3,5))−V₀₁₀₀.

3) For any negative volumes V{α₁, α₂, . . . , α_((i−1)), 1}, adjustmentsmust be made to the φ_(i,j) found in the first step of the GCDEP. Twooptions are possible: the first generally results in slightly loweroverall porosities than the second, and the second generally inrelatively reduced phase separation.

a. Reduce φ_((i−1),(j=x)) to φ′_((i−1),(j=x)), where φ′_((i−1),(j=x)) isdefined as the value such that the smallest volume V{α₁, α₂, . . . ,α_((i−1)),1} of subset “i” is zero, and where particle group “j=x”belongs to both subsets “i” and “i−1”. Using φ′_((i−1)j) in place of theφ_((i−1)j), recalculate all volumes V{α_(n)} for subset “i” and proceedusing the φ′_((i−1)j) in place of the φ_((i−1),j) for all futurecalculations.

b. For the calculations of volumes in subset “i”, but NOT for othervolume calculations, reduce proportionally all φ_((i−1),j) toφ′_((i−1)j), where the φ′_((i−1)j) are defined as the values such thatthe smallest volume V{α₁, α₂, . . . , α_((i−1)),1} is zero, and wherethe particle groups are all groups that belong to subset “i−1” but NOTto subset “i”. Using the φ′_((i−1)j) in place of the φ_((i−1)j),recalculate all volumes V{α_(n)} for subset “i”. If this step is taken,when calculating φ_(i) for subsets “i−1” and “i−2”, the reducedφ′_((i−1)j) must be used.

4) Repeat steps 2) and 3) until volumes for all subsets have beencalculated.

5) Calculate the φ_(j) for each particle group “j” using the volumesV{α_(n)}. In the following formulae, it is assumed that the φ′_(i,j)discussed in Step 3 are substituted for the φ_(i,j) where indicated. Foreach subset “i” beginning with “i=1”, for the particle groups “j” thatdo not belong to any “i” smaller than the subset considered,

a. For particle groups in subsets “i” such that there are no particlegroups belonging only to subset “i”, φ_(j)=φ_(i,j)*ΣV{α₁, α₂, . . . ,α_((i−1)), 0}+φ_((i+1)j)*ΣV{α₁, α₂, . . . , α_((i−1)), 1}, where thefirst sum is over all V{α₁, α₂, . . . , α_((i−1)), α_(i)} for “i” thatdo NOT end in subscript “1” and the second is over all V{α₁, α₂, . . . ,α_((i−1)), α_(i)} that end in the subscript “1”. If there are no subsetswith index greater than “i”, then the value (1−P_(j)) is substituted forφ_((i+1)j) in the second term of the equation. For example, for i=4 andsubsets {2,3,4}₂, {4,5,6}₃, {6,7}₄, {7,8,9}₅,φ₇=φ_(4.7)*(V₀₀₀+V₀₁₀₀+V₀₀₁₀)+φ_(5.7)*(V₀₀₀₁+V₀₁₀₁). If in the previousexample subset 5 were not to be mixed, thenφ₇=φ_(4.7)*(V₀₀₀₀+V₀₁₀₀+V₀₀₁₀)+(1−P₇)*(V₀₀₁+V₀₁₀₁).

b. For particle groups in subsets “i” where there are particle groupsthat belong only to subset “i”, φ_(j)=φ_(ij)*ΣV{α₁, α₂, . . . ,α_((i−1)),0}+φ_(i′j)*ΣV{α₁, α₂, . . . , α_((i−1)), 1}, where the φ_(i′j)are the same as those calculated in Step 2b), i.e., the fractions ofspace occupied by a minimal porosity mixture of all particle groups thatbelong to subset “i−1” but NOT to subset “i”. For example, for i=4 andsubsets {2,3,4}₂, {4,5,6}₃, and {6,7,8,9}₄, φ₇, φ₈, and φ₉ arecalculated as, φ₇=φ_(4,7)*(V₀₀₀₀+V₀₁₀₀+V₀₀₁₀)+φ_(4:7)*(V₀₀₀₁+V₀₁₀₁),φ₈=φ_(4,8)*(V₀₀₀₀+V₀₁₀₀+V₀₀₁₀)+φ_(4:8)*(V₀₀₀₁+V₀₁₀₁), andφ₉−φ_(4,9)*(V₀₀₀₀+V₀₁₀₀+V₀₀₁₀)+φ_(4:9)*(V₀₀₀₁+V₀₁₀₁).

6) When no particle groups are shared between subsets, the super set ofall subsets can be subdivided into sets of subsets within which eachsubset shares particle groups with at least one other particle group inthe set. For example, {1,2}₁, {2,3}₂, {2,5,6}₃, {6,7}₄, {8}₅, can bedivided into three sets, {1,2}₁, {2,3}₂; {4,5,6}₁, {6,7}₂; and {8}₁,each with its own set of volumes for each subset “i”, (V1)·{α_(n)},(V2){α_(n)}, (V3)·{α_(n)}, calculated in the manner described insteps 1) through 6) above. The only difference is that while (V1)₀=1,(V2)₀ is set equal to one minus the total porosity of the particlegroups belonging to set 1, (V3)₀ is set equal to one minus the totalporosity of the particle groups belonging to sets 1 and 2, etc. Usingthe previous example, (V2)0−1−φ1−φ2−φ3, and(V3)₀=1−φ1−φ2−φ3−φ4−φ5−φ6−φ7=V20−φ4−φ5−φ6−φ7.

When more than two particle groups overlap between subsets, amodification to Steps 2a and 2b of the above 6-step technique isnecessary. The modification is straightforward; when calculating volumesfor subset “i”, instead of multiplying by (φ_((i−1)j)/φ_(i,j)) (Step 2a)or (φ_((i−1)′j)/φ_(i,j)) (Step 2b), a general reduction factor takinginto account all overlapping group volume fractions φ_(ij) andφ_((i−1)j) must be used. For example, for groups j1, j2 overlappingbetween subsets “i” and “i−1”, an average of the volume fractions of theoverlapping groups, (φ_((i−1)j1)+φ_((i−1)j2))/(φ_(ij1)+φ_(ij2)) (Step2a) or (φ_((i−1)′j1)+φ_((i−1)′j2))/(φ_(ij1)+φ_(ij2)) (Step 2b), can beemployed in place of the original factor (φ_((i−1)j)/φ_(ij)) (Step 2a)or (φ_((i−1)′j)/φ_(ij)) (Step 2b), respectively. The average can extendto three or more overlapping groups as well. Alternatively, if the largeto small average volume fraction of two overlapping groups is large, forexample, greater than 1,000, then the fact that they both overlap can beignored, and, for the purposes of the volume calculation, it can beassumed that the group with larger average particle size does not belongto subset “i” For example, given groups {1,2,3}_(i) and {2,3,4}₂, thefactor (φ_(1,3)/φ_(2,3)) can be used to calculate V₀₀. Ignoring theoverlap in this case will result in a small decrease in accuracy in thefinal calculated porosity.

For all other Steps in the 6-step technique described above, anylanguage applying to a single overlapping group can be extended to applyto all overlapping groups. For example, if using an average of volumefractions to calculate the reduction factor, then in Step 2c for i=4 andsubsets {3,4}₂, {4,5,6}₃, and {5,6,7}₄, V₀₁₀₁=V₀₁₀*(1−φ_(3,4))−V₀₁₀₀rather than V₀₁₀₁=V₀₁₀*(1−φ_(3,4)−φ_(3,5))−V₀₁₀₀, as groups 5 and 6belong both to subset 3 and subset 4. However, if assuming that onlygroup 6 overlaps, then V₀₁₀₁=V₀₁₀*(1−φ_(3,4)−φ_(3,5))−V₀₁₀₀, as before.

E. Methods for Choosing Particle Groups to Minimize Porosity

In many practical circumstances, available materials in particle formcan be divided into a large number of groups, or there are a largenumber of available groups of particles from which a composite can beformed. For example, in concrete manufacture, there are often as many as10-50 different types of aggregate conveniently located in quarries orother aggregate production facilities near a job site. Or, in additivemanufacturing, powder materials of a certain molecular composition canbe made (or ordered) to meet specific average particle volume andstandard deviation criteria. Generally, when many groups of particleswith average volumes within a small ratio range are available, forexample, 10 groups available with largest to smallest average volumeratio of 1,000,000, using all possible groups to compose a granularcomposite will not yield the lowest porosity. In these cases, it isimportant to choose the right groups in order to achieve low porosities.

Some “rule of thumb” criteria for choosing particle groups include a)choose adjacent groups with large to small average particle volumeratios greater than 25 but less than 2,000, b) where possible, choosegroups with smaller relative standard deviations (e.g., groups witharithmetic standard deviations less than 30% of average particle volume,particularly when the large to small average particle volume ratios ofadjacent particle groups are small), and c) choose groups with higheraverage sphericity, and when mixing methods will not include compactionor vibration steps, with smaller group coefficient of static friction.

The first criteria (a) reflects a balance between reducing porosity andreducing tendency to phase separate. When mixing different particlegroups, both porosity and the tendency to phase separate are stronglydependent on a variety of factors. However, adjacent particle groupswith smaller large to small average particle volume ratios tend to phaseseparate far less easily, but form less dense structures, while particlegroups with larger large to small average volume ratios tend to phaseseparate more easily but form denser structures when not separated. Thesecond criteria (b) reflects the difference between continuous particledistributions and adjacent sufficiently sized particle groups; ifparticle groups are not distinct as described by the GCDEP groupcriteria, these groups tend to pack less densely when mixed. The thirdcriteria (c) reflects a general guideline for mixing multiple groups ofparticles: groups of highly angular, apsherical particles require bothlarger large to small average particle volume ratios and particularlycareful preparation to mix into composites with reduced porosities; itis more simple to achieve low porosities when using spherical particleswith low coefficients of static friction.

Balancing granular composite porosity with other desired physicalcharacteristics, including but not limited to relative viscosity andtendency to phase separate, requires substantial knowledge of particlegroups' physical characteristics, physical interactions, sizedistributions, particle geometry, method of mixing, and potentiallyother factors as well. However, given substantial knowledge of some orall of these critical factors, the choice of which particle groups tomix in order to achieve desired results (including, for example, findingthe minimal porosity structure that won't phase separate under certainapplications given a fixer upper limit on relative viscosity) can bemade by considering particle group mixing as an optimization problem.

Provided a superset of particle groups that conform to the GCDEP groupcriteria, the first step of the GCDEP process can be used in conjunctionwith optimization methods to minimize the porosity of a granularcomposite composed of a subset of the particle groups. This approach isaided by the division of the superset of particle groups into subsets,an example of which is illustrated in FIG. 6, and by knowledge of theporosities of the subsets, mixed as they will be mixed for theapplication of the granular composite, as a function of relative volumesof each group in the mixture. For a given subset “i” containing N_(i)particle groups “j” these porosities can be written as P_(i)(φ_(i,j1),φ_(ij2) . . . φ_(ijN)) for volume fractions φ_(i,jn).

Given these porosity functions P_(i)(φ_(ij1), φ_(ij2) . . . φ_(ijN)),the problem of choosing groups becomes a nonlinear programming problemin a number of variables equal to Σ_(i)(N_(i)−1), where the sum is overall subsets “i” and the “minus 1” term is due to the fact that given theporosity P_(i) and “N_(i)−1” of φ_(ijn), the last of the φ_(i,nj) can bedetermined. Those familiar in the art can construct the objectivefunction of the programming problem using the 6-step approximationtechnique described above or any other accurate subset combinationtechnique that is devised. Standard constraints can be included in theproblem, for example, φ_(inj)>=0, or to reduce phase separation, anyφ_(ijn) can be set so that it is greater than, by any amount, theφ_(ijn) corresponding to the minimum in porosity. Constraints can alsobe set to incorporate Step 3 of the 6-step approximation technique. Anyaccurate programming technique, including but not limited to AugmentedLagrangian, Quasi-Newton, Barrier, Conjugate Gradient, etc., can be usedon its own or in conjunction with other methods to obtain reducedporosity solutions. In using these methods, for large numbers ofparticle groups, it is likely that many of the volume fractions φ_(ijn)will be zero whenever j=jn, indicating that group “j” is not included inthe final composite. This non-zero volume fractions indicate the groupsthat should be included in the final composite.

Partial knowledge of P_(i)(φ_(ij1), φ_(ij2) . . . φ_(ijN)) over therange of interest in the φ_(ijn) for each subset often requirestime-consuming methods to obtain. However, this knowledge is notgenerally necessary for the optimization method. Rather, it is morepractical when the number of particle groups in a subset “i” is large toobtain partial subset porosity functions P_(j1j2)(φ_(ij1), φ_(ij2)) forthe mixtures of pairs of particle groups where the large to smallaverage volume ratio of the pair is less than 10,000, and the functionsP_(j1,j2,j3)(φ_(i,j1), φ_(ij2), φ_(ij3)) for mixtures of triplets ofparticle groups where the large to small average volume ratio of thepair is less than 10,000. Groups of four, five, six, etc. particlegroups under the same average volume ratio criteria can also beemployed. However, given particularly large supersets of particlegroups, the “rule of thumb” criteria previously mentioned can be used toreduce the number of required porosity functions.

Given the partial subset porosity functions P_(j1j2), . . . (φ_(ij1),φ_(ij2) . . . ) and the subset segregation procedure, an example ofwhich is given in FIG. 6, those skilled in the art can construct anobjective function for a programming problem using the 6-stepapproximation technique described earlier. Constraints can be added inthe same fashion as when using the full P_(i)(φ_(ij1), φ_(ij2) . . .φ_(ijN)) functions. Additionally, this approach requires using in theobjective function groups of particles corresponding only to sets of theP_(j1j2), . . . (φ_(ij1), φ_(ij2) . . . ) such that each P_(j1j2) . . .(φ_(ij1), φ_(ij2) . . . ) forms a subset via the subset segregationprocedure illustrated in FIG. 6. Consequently, it may be necessary toconsider and/or solve several different programming problems, each withobjective function and constraints corresponding to different sets ofP_(j1,j2), . . . (φ_(ij1), φ_(ij2) . . . ). However, these programmingproblems will be significantly more time-efficient to solve given thereduction in variables in each problem due to the simpler forms of theP_(j1j2), . . . (φ_(ij1), φ_(ij2) . . . ).

Finally, the complete or partial porosity functions P_(i)(φ_(ij1),φ_(ij2) . . . φ_(ijN)) and P_(j1,j2), . . . (φ_(ij1), φ_(ij2) . . . )are not necessary in order to choose, from a large superset of groups,which groups to mix to achieve a granular composite with substantiallyreduced porosity. Given a single point from each porosity functionP_(i)(φ_(ij1), φ_(ij2) . . . φ_(ijN)), or a single point from each of aset of chosen P_(j1j2), . . . (φ_(ij1), φ_(ij2) . . . ), where each ofthe set of P_(j1j2), . . . (φ_(ij1), φ_(ij2) . . . ) forms a subset viathe subset segregation procedure illustrated in FIG. 6, the 6-stepprocedure described in Subsection B of the previous Section can be usedto obtain a reduced porosity mixture. In particular, if the points thatare given are at the minima in porosity or near to the minima inporosity of the porosity functions, the 6-step technique described inSubsection B of the previous section will yield a substantiallyreduced-porosity mixture.

DESCRIPTION OF PREFERRED EMBODIMENTS

As noted above, in one embodiment, the present invention contemplatesadditive manufacturing, including but not limited to Selective lasersintering (SLS). SLS is a technique used for the production of prototypemodels and functional components. SLS uses lasers as its power source tosinter powdered material, binding it together to create a solidstructure. The aim of the laser beam is scanned over a layer of powderand the beam is switched on to sinter the powder or a portion of thepowder. Powder is applied and successive layers sintered until acompleted part is formed. The powder can comprise plastic, metal,ceramic, carbide, glass, and polymer substances (as well as combinationsthereof).

Selective Laser Sintering and Direct Metal Laser Sintering areessentially the same thing, with SLS used to refer to the process asapplied to a variety of materials—plastics, glass, ceramics—whereas DMLSoften refers to the process as applied to metal alloys. But what setssintering apart from melting or “Cusing” is that the sintering processesdo not fully melt the powder, but heats it to the point that the powdercan fuse together on a molecular level.

Selective Laser Melting, or SLM (often called Direct Metal LaserMetaling, or DMLM), on the other hand, uses the laser to achieve a fullmelt. In this case, the powder is not merely fused together, but isactually melted into a homogenous part. Melting is typically useful fora monomaterial (pure titanium or a single alloy such as Ti 6-4), asthere's just one melting point. By contrast, it is currently typicalwhen working with multiple metals, alloys, or combinations of alloys andother materials such as plastics, ceramics, polymers, carbides orglasses, to use SLS or DMLS.

The present invention contemplates using increase porosity powders withSLS and DMLS. In addition, such powders are herein contemplated for SLMand DMLM.

Generally, in the field of additive manufacturing using composites suchas powders, little work has been done with respect to optimizing thestructures of the composites. It is not known, even among experts in thefield, that composites with greatly reduced porosity can be produced,where those composites do not easily phase separate and are stillmalleable enough to be used in standard applications. Metal and ceramicpowders employed in selective laser melting and selective lasersintering processes generally exhibit porosities of about 30% to 50%. Inthe scarce academic literature available on the subject of the effect ofpowder porosity on laser sintering and laser melting processes, powderswith porosities of as low as 25% have been produced. E. O. Olakanmi, K.W. Dalgarno, and R. F. Cochrane, Laser sintering of blended Al—Sipowders, Rapid Prototyping Journal 18, 109-119 (2012). The reasons forthis small body of work involving lower porosity powders include, butare not limited to, that a) the extent of the advantages of usingpowders with porosities less than about 25% are not known, and b) alow-cost production process that produces powders that do not easilyphase separate, maintain the malleability to be processed easily (i.e.,have low enough viscosity), and exhibit porosities less than about 25%,is not known.

As a general process improvement to additive manufacturing processesusing metal, ceramic, cermet, glass, carbide, or other high-meltingtemperature powders along with focused melting by laser or otherprocess, the use and advantages of employing low-porosity powders,defined as powders with porosities less than or equal to 20%, aredescribed. These low-porosity powders can be produced by either thegranular composite density enhancement process or by some other process(this distinction is discussed in detail below). The use of low-porositypowders can be undertaken with powder particles composed of metals,cermets, ceramics, glasses, polymers, and alloys of any elemental ormolecular composition. In particular, the threshold value of 20%porosity is set for low porosity powders to reflect the upper bound ofthe porosities of powders used and capable of being used in currentadditive manufacturing processes.

Additive manufacturing as an industry and the research community have attimes recognized some benefits to decreasing the porosity of powders.This is evident in the sometimes-employed “compaction step” undertakenusing rollers or vibrating hoppers in the laser melting and lasersintering processes to decrease porosity of powders from about 35-50% toabout 25-35% before melting or sintering using laser energy. However, inthe few cases where it has been investigated, decreasing porosity beyondthese amounts is often not seen as desirable. For example, one of thefew studies that includes data on some of the effects of reducing powderporosity in a laser sintering process on laser sintered parts, statesthat “Results from [other] studies have not been able to completelydefine what direct consequences the nature of particle packingarrangements has on the processing conditions, densification, andmicrostructure of laser sintered components.” See E. O. Olakanmi et at(cited above). Olakanmi et al studied metal powders ranging in porosityfrom roughly 28% to 37%. Though some of their lower-porosity powdersproduce greater density printed parts, which is desirable, in othercases, they found “no strong correlation” between the density of printedparts and the porosity of the powders. They concluded that it is notclear whether decreasing porosity beyond the experimental parameters oftheir work can be beneficial to the laser sintering process.

In direct contrast to these conclusions, the present inventors haveidentified a broad range of advantages to the laser melting and lasersintering processes due to employing low-porosity powders. A discussionof these advantages follows.

Reductions in porosity increase overall absorption of laser energy inpowders, particularly as powder layers become thinner and thinner, forexample, for layers less than 100 nm thick. The increase in absorptionmeans that mom laser energy is used for melting particles, rather thanenergy being scattered into the atmosphere of the printer chamber,making printing with low-porosity powders more efficient. As lasermelting and laser sintering processes require expensive high-powerlasers, the ability to reduce required laser power can result insignificant cost savings in the laser component of printers. Forexample, in a laser melting process, comparing the absorption of astandard Titanium-based alloy powder of about 40% porosity with skindepth of about 65 μm to a low-porosity Titanium-based alloy powder ofabout 10% porosity with skin depth of about 20 μm, the low-porositypowder will absorb approximately 2.5× as much laser energy, therebyallowing the use of a laser that outputs 60% less power.

Reductions in porosity also decrease lateral scattering of laser energyin laser heating of powders, meaning that heating is more focused. Thisleads to a more uniform, more controlled melting or sintering process.

Reductions in porosity exponentially increase the thermal conductivityof powders. This is due in part to the increased number of conductionpathways present in lower-porosity powders, which have more contactsbetween particles. The increase in thermal conductivity from a 40%porosity powder to a 5% porosity powder can be as much as 10× to 50×,and this change can result in a multitude of improvements for lasermelting and laser sintering processes.

One improvement resulting from this increase in thermal conductivity isa greatly reduced temperature gradient present over the powder layer asit heats, melts, and re-solidifies. This reduced gradient can be aslarge or larger than 250,000 degrees Kelvin per centimeter. A largertemperature gradient means more thermal expansion in some areas of theliquid, and therefore more volume change from liquid to solid, leadingto uneven re-solidification in the sense of more grains and grainboundaries (including cracks) in the metal solids. Powders withincreased thermal conductivity therefore can resolidify with fewer grainboundaries and cracks. This leads to greater strength of the solid andgreater durability under stressed, high temperature, and corrosionconditions, as well as improved part surface structure due to more evenheat flow and therefore melting.

The reduced temperature gradient induced by higher powder thermalconductivity also extends into the metal material below the powder. Thismeans that more of the previously printed layers will be heated to agreater temperature when the powder has relatively higher thermalconductivity, which means more annealing of the metal, leading to areduction in grain boundaries and a generally more uniform finished-partmicrostructure. This also leads to a reduced tendency of the liquidmetal to “ball” on the surface, which is undesirable as balling leads topore formation in the final printed solid, because interface energiesare reduced under lower temperature gradients across the interface.

A relatively higher powder thermal conductivity also leads to relativelylower maximum temperatures exhibited during a laser melting or lasersintering process. The difference in maximum temperature can exceed 1000degrees Kelvin, where the highest temperatures occur at the powdersurface (which liquefies and sometimes boils). Relatively higher maximumtemperatures mean more oxidation, which is undesirable due tooxidation-induced balling and to the introduction of oxide impurities.Relatively higher maximum temperatures also mean more “splatter” of thesurface liquid, resulting in a rougher surface and possibly micro ormacro pore formation. Pore formation leads to weaker, less durable, lesscorrosion-resistant solid parts. Relatively higher maximum temperaturesadditionally mean, in alloys, more phase separation, for example, ofCarbon migrating to the surface in liquid stainless steel. Phaseseparation is undesirable, as it leads to weaker structure and evendelamination of layers.

Reductions in porosity also decrease the amount of super-heated gaspresent in the spaces between particles in a powder structure. This isimportant, as that gas must escape during the melting andre-solidification process. Gas escape during solidification can lead tomicro and macro pore formation, as well as grain boundary formation.

Due to the advantages including but not limited to those discussedpreviously, and due to both the inconclusive nature of past studiesinvestigating the effects of employing relatively denser powders inadditive manufacturing processes involving powders and to the previousinability in general practice of producing workable (meaning low enoughviscosity and minimal phase separation) powders with porosities of 20%or less, the present inventors propose, as a general process improvementto additive manufacturing processes where composite layers are sinteredor melted in succession to form final products, the use of powdersexhibiting porosities of 20% or less. A method, the granular compositedensity enhancement process, to produce such powders from a range ofdifferent materials including but not limited to pure metals, alloys,ceramics, cermets, and glasses, is described above. Specific examples ofhow to produce workable powders with porosities of 20% or less using thegranular composite density enhancement process are given in theExperimental section. Examples of how to produce workable powders withporosities of 20% or less using other processes are also found in theExperimental section.

Employing a powdered metal ink for a Direct Metal Laser Sintering (DMLS)or Selective Laser Melting (SLM) process, print speed can bedramatically enhanced using an ink produced according to the granularcomposite density enhancement process described herein. It is useful tocompare (by way of example) an “enhanced ink” to a standard metal powderconsisting of approximately same-size, roughly spherical particles.Assuming that the melt time of the ink is the limiting factor in printspeed, a minimum 50-100 times increase in print speed is expected due tothe increased thermal conductivity and lower porosity of the enhancedink. These calculations assume a neutral background gas of Ni or Ar atapproximately 1 atmosphere of pressure and are valid for various metal,ceramic, carbide, and other inks including Ti, Ti alloys, StainlessSteel, Copper, Nickel-based superalloys, Aluminum Oxide, TungstenCarbide, and any other material with bulk thermal conductivity greaterthan 1 W/m*K. At lower pressures and for materials with higher bulkthermal conductivities, the increase is more pronounced. The increase inprint speed is accompanied by significant increases in the bulk, shearand Young's moduli (i.e., increased mechanical strength), and inelectrical and thermal conductivity. Additionally, significantimprovement in the reproducibility of high-quality printed products, dueto reduced void fraction and defects, is expected in the resultantprinted product.

When we compare (by way of example) an “enhanced ink” to a metal powderconsisting of a mixture of two sizes of roughly spherical particles, thelarger panicles are assumed to be roughly 100 times larger than thesmaller, by volume. Assuming that the melt time of the ink is thelimiting factor in print speed, a 5-25 times increase in print speed isexpected due to the increased thermal conductivity and lower porosity ofthe enhanced ink. These calculations assume a neutral background gas ofN₂ or Ar at approximately 1 atmosphere of pressure and are valid forvarious metal inks including Ti, Ti alloys, Stainless Steel, Copper, andany other metallic element or alloy with bulk thermal conductivitygreater than 1 W/m*K. At lower pressures and for metals with higher bulkthermal conductivities, the increase is more pronounced. The increase inprint speed is accompanied by significant increases in the bulk, shearand Young's moduli (i.e., increased mechanical strength), and inelectrical and thermal conductivity. Additionally, significantimprovement in the reproducibility of high-quality printed products, dueto reduced void fraction and defects, is expected in the resultantprinted product. The increases in mechanical strength, conductivities,and reproducibility are expected to be somewhat less extreme than thoseobtained when comparing to a generic powder ink.

In describing the enhanced ink, the most important point to remember isthat different microstructures, even at similar porosities (porosity isinterchangeable with density when a single alloy or element isspecified), can result in different characteristics, includingviscosities, conductivities (related to print speed), and mechanicalstrengths.

EXPERIMENTAL Example 1

This is an example of the production of a dense powder for additivemanufacturing purposes without using the GCDEP. Ti 6-4 (Ti-6Al-4V, ortitanium containing (by weight) about 6% aluminum, 4% vanadium, and someminimal trace elements, including but not limited to iron and oxygen).In this example, two groups of Ti 6-4 particles of commercial puritywith a large difference in averages particle sizes are used to produce acompacted metal powder with porosity of approximately 18%. The firstgroup of particles exhibit approximately normally distributed volumeswith average particle volume of about 8,000 μm3 (effective diameter ofabout 25 μm) and a small diameter standard deviation of 5 μm. Thevolumes of the second group are approximately log-normally distributedaccording to effective diameters (meaning an approximately normallydistributed passing curve, and also volume probability density function,as a function of the logarithm of particle effective diameter orlogarithm of particle volume), with an average particle volume of 0.5μm3 (effective diameter of about 1 μm), and a diameter standarddeviation of 0.5 μm. Both groups of particles are highly-spherical(sphericity >0.95), and the coefficient of frictional interactionbetween particles within the same group is about 0.28, indicating thepresence of a thin (<5 nm thick) natural oxide layer surrounding theparticles. The un-compacted packing traction (traction or space coveredby the particles) of the first group is 0.60, and that of the secondgroup is 0.595, consistent with the sphericity and coefficients offriction of the groups. When thoroughly mixed without vibration inlarger group to smaller group mass fractions of 71%:29%, porosity of thebulk material will be approximately 18%. Exact porosity is highlydependent on mixing method, as these particles will have a tendency tophase separate, for example, when vibrated. The phase separated mixturewill exhibit a porosity of about 40%. To achieve 18% porosity, thoroughup-down mixing with minimal vibration and a compaction step arenecessary. For example, in additive manufacturing, if a 100 μm layer ofthe mixed power is deposited using a roller, this roller can pass asecond time over the deposited layer except at a slightly lower height,thereby compacting the mixture. In this case, the surface of the layerwill exhibit higher porosity than the bulk, but only to ˜10-20 μm ofdepth.

Example 2

This is an example of the production of a dense powder for additivemanufacturing purposes without using the GCDEP. 316 Stainless Steel(approximately 16.5% carbon, 12% chromium, 3% nickel, 1.4% molybdenum,0.8% silicon, and trace phosphorus, sulfur, other elements). In thisexample, stainless steel particles of commercial purity in a singlegroup exhibiting a continuous, approximately log-normally distributedsize (diameter) probability density function with average particleeffective diameter of about 50 nm (effective volume of 65,000 nm3),sphericity of 0.86, and frictional coefficient of 0.52, are used toproduce a powder with porosity of about 20%. To accomplish this, thegeometric standard deviation of the particle group must be about 5.5 μm,meaning that 5% of the total volume of particles will be greater than 1mm in diameter (and 40% greater than 100 μm in diameter). In this case,for additive manufacturing, a vibration step will be advantageous inreducing porosity, as phase separation will not likely be a concern.However, laser processing of layers 1 mm thick is complicated, as evenparticle melting (with minimal evaporation) is inhibited by the amount,of time required for the bottom of the layer, 1 mm from the point wherethe laser first strikes the surface, to heat to melting temperatures.Additionally, the smaller nanometer-scale particles, having a muchhigher ratio of surface area to volume, are more subject to forces,including but not limited to electrostatic, Casimir and Van der Waalsforces, that cause them to “stick” to each other and larger particles.To counter electrostatic forces, an environment with minimal residualcharge (a “static-free” environment) is important, and an additionalcompaction step likely necessary, to achieve 20% porosity.

Example 3

This is an example of the production of a dense granular composite usingthe GCDEP. Two groups of flint glass (also called soda-lime) beads withhigh sphericity >0.98 and low static coefficient of friction <0.05 areto be mixed to form a macroscopic filter. The larger group of beads are10 mm in diameter and the smaller are 2 mm in diameter. The beadmixtures are highly uniform, i.e., the standard deviation of the volumeprobability density function of each group is approximately zero.Referring to Table 1, a catalogue of various critical porosity valuesfor simulated mixtures of frictionless spheres at various small to largeaverage sphere diameter (and volume) ratios (simulations conducted usingthe TJ algorithm), the minimum porosity of 21.6% is found to occur at arelative volume fraction of small spheres of 20.6%. No compaction orvibration step is necessary to achieve this porosity; in fact, excessvibration will result in phase separation of the beads at these sizeratios. Upon mixing, the minimum in porosity was found to be 2.1.4%,occurring within 0.9% of the simulated value, at a relative volumefraction of small spheres of 20.1%, within 2.4% of the simulated value.The slightly lower porosity found at somewhat smaller small sphererelative volume fraction was likely the result of some ordering of thelarger spheres occurring during the mixing. FIG. 7 contains imagesdisplaying the experimental apparati of these tests. The beaker in theimages contains a mixture of a relative volume fractions of about 17.5%2 mm beads and 82.5% 10 mm beads with porosity of 22.2%. This image isnot captured at the minimum porosity structure found for this diameterratio of beads.

For mixtures of 1 mm and 10 mm beads exhibiting the same physicalcharacteristics as the beads above, a minimum porosity of 17.7% wasfound at 25.0% relative volume fraction of small beads; this is within0.6% of the minimum porosity as predicted by the simulations, with theminimum occurring within 0.4% of the critical relative small spherevolume fraction (data shown in Table 1).

For mixtures of 1 mm and 3 mm beads exhibiting the same physicalcharacteristics as the beads above, a minimum porosity of 28.8% wasfound at relative small sphere volume fraction of 24.5%, which is within0.7% of the porosity and the minimum within 0.4% of the relative smallsphere volume fraction predicted by simulations (data shown in Table 1).

Example 4

This is an example of the production of a dense granular composite usingthe GCDEP. Two groups of highly spherical (sphericity ×0.95), very lowfriction (static coefficient of friction equal to 0.07, kineticcoefficient of 0.03) Tungsten Disulfide particles are to be mixed tominimize porosity. The first group exhibits a normal distribution ofdiameters about a mean size of 10.1 μm with a standard deviation of 0.9μm, and the second, also normally distributed, a mean size of 1.4 μmwith a standard deviation of 0.14 μm. As the coefficients of frictionare very low, the sphericity very near 1, and the standard deviations asa percentage of mean particle size very small, these particle groups canbe considered as frictionless spheres. Referring to Table 1, the mixturethat minimizes porosity is determined to occur at a relative volumefraction of 22.4% smaller particles, with a porosity of 18.7%. Thesevalues are determined by linear extrapolation to a diameter ratio of1.4/10.1=0.137 from the values given in the table for small to largediameter ratios of 0.10 and 0.15. Due to low coefficients of friction,vibrating or compacting is not necessary to achieve low porosities, butto negate any electrostatic forces arising in the mixing procedurewithout inducing phase separation, a compaction step could sometimes benecessary.

Example 5

This is an example of the production of a dense granular composite usingthe GCDEP. Three groups of highly aspherical, high coefficient offriction particles, consisting of crushed rock and sand, are mixed witha paste, consisting of cement water, to make concrete. In this example,the amount of cement required to formulate the concrete is minimizedsuch that the concrete is still “workable”, meaning that it will stillflow and, after compacting, fill spaces around rebar such that nomacroscopic voids remain. The largest group of particles consists ofhighly elongated crushed granite with average roundness of 0.21, density2.79 g/cm³, average sphericity of 0.56, and coefficient of staticfriction of 0.92. The particles have been sieved between 4.00 mm and4.18 mm sieves to yield a number average size of 4.09 mm with a roughlyuniform size distribution from 4.00 mm to 4.18 mm. The porosity of thisgroup of particles (on its own) is P1=52%. The second group of particlesconsists of natural sand of average roundness 0.54, density 2.66 g/cm3,average sphericity of 0.76, and coefficient of static friction of 0.70.The particles have been sieved between 1.00 mm and 1.25 mm sieves toyield a number average size of 1.13 mm with a roughly uniform sizedistribution from 1.00 mm to 1.25 mm. The third group, also naturalsand, exhibits similar characteristics to the second, except it has beensieved through 0.15 mm and 0.20 mm sieves, yielding a roughly uniformparticle distribution from 0.15 mm to 0.20 mm with average size of 0.18mm. The second group exhibits a porosity (on its own) of P₂45%, and thethird group P_(j)44%. The fourth group of particles consists of Portlandgrey cement; it exhibits an average roundness of 0.78, density of 3.15g/cm, average sphericity of 0.88, and coefficient of static friction of0.38. Its volume probability density function is roughly log-normallydistributed around an average particle volume of 33,500 μm³ (effectivelinear size (diameter) of 40 μm) with arithmetic standard deviation ofabout 97,500 μm³ (57 μm in units of linear size), meaning that 97% ofthe volume of cement particles have a linear size smaller than 150 μm(69% smaller than 75 μm). The porosity of the cement (on its own) is30%. However, since the size distribution and average particle volume ofthe cement will change (particles will become smaller) when the cementreacts chemically with water, the size distribution and average particlevolume are in this case ignored. This is possible given the reactionwith water, but were the smallest sand particles more similar in size tothe cement particles, the cement average particle volume and sizedistribution could not necessarily be discounted.

Dividing the three groups of particles into subsets using the methoddepicted in FIG. 6 yields two subsets of particle groups, {1,2}₁ and{2,3}₂. In this case, for each subset, the relative volume fraction ofsmaller group particles at which minimal porosity occurs is determinedexperimentally using the experimental method described in Subsection Babove. As previously stated, no vibration or compaction steps were takenin measuring volume fractions. For subset 1, minimal porosity is 44% andoccurs at a relative volume fraction of group 2 particles of 40%. Forsubset 2, minimal porosity is 26% and occurs at a relative volumefraction of group 3 particles of 24%. Using the second step of theGCDEP, described in Subsection B above, the volume fraction of group 1particles is found to be φ₁=0.56*0.60=0.336, and the volume fraction ofgroup 2 particles is found to be φ₂=0.56*0.40=0.224. The quantityV₀₀=0.398 and V₀₁=0.266, yielding φ_(e)=V₀₀·φ_(2,3)+V₀₁·(1−P₃=0.220. Thevolume fraction of group 3 particles is found to be(1−0.364−0.358)·0.56=0.156. The porosity of the aggregate mixture is22.2%.

To this mixture of three groups of aggregates is added a paste of cementin water. There must be at least enough paste to fill the porositybetween aggregates; however, generally, more paste is added in order toallow the wet concrete mixture to flow. Cement and water are usuallycombined in a pre-determined mass ratio, since the properties of thecement paste depends on the amount of water mixed in. However, extrawater must be added to account for the water absorption of theaggregates. Each group 1, 2, and 3 of aggregates absorb, respectively,1.1%, 1.5%, and 1.6% of their masses in water. In this case, cement wasadded in a mass fraction of 11.9% of the total mixture mass, with 0.45grams of water added for every 1 gram of cement, and an additional 0.09grams of water per gram of cement to account for aggregate absorption.If the cement and water were assumed to maintain their densities uponmixing (i.e., the chemical reactions were ignored), then the massfraction of cement required (with water added in a mass fraction ofwater:cement, 0.54:1) to fill the porosity in the aggregate would havebeen 10.0%. However, with the improved viscosity afforded by addingextra cement, the final mass fractions of components were36.1%:23.0%:22.5%:11.9%:6.5% group 1:group 2:group 3:cement:water. Thisrepresents a total of 74.6% aggregate, 9.4% cement, and 16.0% water byvolume.

Example 6

This is an example of the production of a dense granular composite usingthe GCDEP. Three groups of particles, the first two structural materialsconsisting of spherical cast tungsten carbide (WC) powder, and the thirda binding material consisting of cobalt (Co) powder, are mixed to form acomposite powder for selective laser sintering. The WC particles arehighly spherical (average sphericity ×0.97) and exhibit low staticcoefficient of friction of 0.08 due to the addition of small amounts oflubricant, which will burn off at low temperatures (<500 degrees C.)during sintering. The first group of particles has uniform sizedistribution from 111 μm to 118 μm, with average particle volume of 114μm. The second group also has uniform size distribution, from 20 μm to26 μm, with average particle volume of 23 μm. Considering the highsphericity and low coefficients of friction in the first two particlegroups, these groups can be approximated as frictionless spheres. TheCobalt particles are rounded and somewhat aspherical, with sphericity of0.84 and coefficient of static friction of 0.37. Their volumes areroughly log-normally distributed about an average of 0.4 μm3 (effectivediameter of 0.9 μm) with arithmetic standard deviation of 0.25 μm3(effective diameter standard deviation of about 0.78 μm). The Cobaltparticles exhibit an uncompacted porosity of 31%. Dividing the threegroups into subsets via the method illustrated in FIG. 6 yields twosubsets {1,2}₁ and {3}₂, where the third group is small enough to beassumed to fit in the void space left by the two larger groups.

In this case, the thickness of powder layers to be sintered at each stepin the additive manufacturing process is 250 μm-350 μm. To minimizevariations in layer surface thickness, minimal porosity is not sought inmixing the first two sets of particles, but rather a mixture thatincludes 35% (by volume) 23 μm particles, thereby diluting the largerparticles and reducing lateral surface thickness variation. Referring toTable 2, a catalogue of porosity values at different small sphererelative volume fractions for simulated mixtures of frictionless spheresat various small to large average sphere diameter ratios, theuncompacted porosity of the 65% group 1.35% group 2 mixture isdetermined to be 24%. In an ideal fabrication process, the melted Cobaltwould percolate and completely fill the void space within the WCmixture. At an uncompacted Cobalt powder porosity of 31%, this wouldrequire volume ratios of 0.65:0.35:0.35 group 1:group 2:group 3particles. Written in mass fractions, this is 55%:30%:15% group 1:group2:group 3 particles. Due to the small average particle size of Cobaltparticles relative to the group 2 WC particles, a compaction step mightbe necessary before sintering in order to minimize phase separation ofthe Cobalt from the WC particles.

It should be noted that in most fabrication processes, in part due tothe different densities of liquid and solid Cobalt, the materialproduced after laser sintering would be porous, and could thereforerequire less mass of Cobalt than used in this example.

Example 7

This is an example of the production of a dense granular composite usingthe GCDEP. Four groups of Ti 6-4 particles, each comprising about 89.5%titanium, 6% aluminum, 4% vanadium, 0.3% Iron and 0.2% Oxygen, as wellas trace elements, are to be mixed to form a powder for a laser meltingadditive manufacturing process. The first group of particles exhibitsaverage sphericity of 0.97, a coefficient of static friction of 0.25,and is distributed roughly uniformly in particle volume from 240 μm³ to380 μm³ (7.7 μm to 9.0 μmin effective diameter) with average particlevolume of 302 μm³ (8.3 μm in effective diameter). The second groupexhibits similar average sphericity and coefficient of static friction,and is distributed uniformly in particle volume from 2.5 μm³ to 7.1 μm³(1.7 μm to 2.4 μm in effective diameter) with average particle volume of4.8 μm³. The third particle group exhibits average sphericity of 0.91, acoefficient of static friction of 0.37, and is distributed log-normallyin its volume probability density function with average particle volumeof 0.075 μm³ (effective diameter of 525 nm) and standard deviation ofabout 0.015 μm³. The fourth particle group exhibits average sphericityof 0.87, a coefficient of static friction of 0.42, and is distributedlog-normally in its volume probability density function with averageparticle volume of 0.000697 μm³ (effective diameter of 110 nm) andstandard deviation of about 0.000290 μm³. Dividing the four groups intosubsets via the method illustrated in FIG. 6 yields two subsets,{1,2,3}₂ and {3,4}₂.

With the size distributions, sphericities and coefficients of staticfriction known, simulations using the TJ algorithm yield a porosityminima and the relative volume fractions at which they occur for thesubsets. Since the composite will be compacted before processing,compaction is also taken into account in the simulation, which yieldsfor the first subset a porosity of 12.3% at relative volume fractions of65.8%:19.7%:14.5%, and for the second subset a porosity of 23.4%occurring at relative volume fraction of 22.3% group 4 particles, wherethe group 4 particles on their own exhibit a porosity P4=38.8%. Thesecond step of the GCDEP, as discussed in Subsection B above, is appliedto yield Y₀₀=0.213 and V₀₁=0.0457, and with φ₁=0.578, φ₂=0.172, andφ₃=0.127, yields V₀₀=0.213 and V₀₁=0.0457, This gives a final porosityof 6.4% with relative volume fractions of 61.7%:18.4%:13.6%:6.3% group1:group 2:group 3:group 4 particles.

Example 8

This is an example of the production of a dense granular composite usingthe GCDEP. Seven groups of alumina (Al₂O₃) powders of high purity(>99.0%) are to be mixed, compressed at high pressure (200 MPa), andsolid-state sintered at 1550 degrees C. for use as granular armor. Theparticle groups have the following properties:

1. The to group consists of cylinders (sphericity of about 0.87) 12.2 mmin length and 8.6 mm diameter, with roughly uniformly distributedvolumes from 680 to 740 mm³ (average volume of 710 mm³). Theircoefficient of friction is 0.34.

2. The second group consists of angular, aspherical particles(sphericity of 0.72) with roughly uniformly distributed volumes from19.5 mm3 to 22.0 mm³ (average volume of 21.8 mm³, effective averagediameter of 3.4 mm). Their coefficient of friction is 0.74.

3. The third group consists of angular, aspherical particles (sphericityof 0.75) with roughly uniformly distributed volumes from 0.55 mm3 to 0.9mm³ (average volume of 0.73 mm3, effective average diameter of 1.1 mm),Their coefficient of friction is 0.59.

4. The fourth group consists of highly spherical particles (sphericityof 0.98) with normally distributed volumes averaging 0.0016 mm3 andstandard deviation of 0.0004 mm³ (effective average diameter of 145 μm).Their coefficient of friction is 0.17.

5. The fifth group consists of highly spherical particles (sphericity of0.97) with roughly uniformly distributed volumes from 5100 μm³ to 6050μm³ (average volume of 5580 μm³, effective average diameter of 22.0 μm).Their coefficient of friction is 0.21.

6. The sixth group consists of rounded, somewhat aspherical particles(sphericity of 0.88) with log-normally distributed volumes averaging 69μm³ with a standard deviation of 5.2 μm³ (effective average diameter of5.1 μm). Their coefficient of friction is 0.38.

7. The seventh group consists of somewhat rounded, aspherical particles(sphericity of 0.84) with log-normally distributed volumes averaging0.27 μm³ with a standard deviation of 0.05 μm³ (effective averagediameter of 0.8 μm). Their coefficient of friction is 0.47.

Dividing the seven particle groups into subsets via the methodillustrated in FIG. 6 yields five subsets, {1,2,3}₁, {3,4}₂, {4,5}₃,{5,6}₄, and {6,7}₅. For each subset, to ensure that daring compressionthere are no large voids, the relative volume of smaller spheres ischosen to be about 20% larger than the volume that corresponds to theminimum in porosity. Simulation using the TJ algorithm is used todetermine the desired relative volumes for the first subset, yielding aminimum compacted porosity of 14.5%; however, for the volume fractionsemployed, compacted porosity 16.9% at relative volumes of46.4%:25.9%:27.3% group 1:group 2:group 3 particles. For the secondsubset, experimentation is used to determine a curve of porosity versusgroup 4 relative volume fraction of particles. The result is a compactedporosity minimum of 18.8% occurring at 28% group 4 particles; however,for the mixture of subsets, 33.5% group 4 particles are used, giving aporosity of 19.7%. For the third subset, under compaction, both particlegroups of particles can be approximated as frictionless spheres. Usingthe lookup Table 2 at small to large diameter ratio of 0.15 yields, vialinear interpolation, a porosity of 19.9% at 27% relative volumefraction of group 5 particles. For the fourth and fifth subsets,simulation using the TJ algorithm is employed, again using values forrelative volume fractions of smaller particles that are about 20% largerthan the values for which the porosity minimum occurs. This approachesyields porosities of 25.5% and 21.8% occurring at relative volumefractions of smaller particles of 29.5% and 25.2%, respectively. Thecompacted porosity P₇ of particle group 7 on its own was simulated to be34.2%. Experiment placed P₇ at 33.9%, which was the value used.

Using the six-step method described herein, the subsets are combined.The following values are calculated:

1. V₀=1, φ₁=0.386, φ₂=0.215, φ₃=0.227

2. V₀₀=0.425, V₀₁=−0.0261. According to Step 3b) of the subsetcombination technique, φ₁ is reduced to 0.369 and φ₂ to 0.206, yieldingV₀₁=0. The value φ₄=0.114.

3. V₀₀₀₀=0.080, V₀₀₀₁=0.0070, V₀₁₀₀=0, V₀₁₀₁=0, V₀₀₁₀=0.0026. The valueφ₆=0.0480.

4. V₀₀₀₀=0.080, V₀₀₀₁=0.00070, V₀₁₀₀=0, V₀₁₀₁=0, V₀₀₁₀=0.0026. The valueφ₆=0.0480.

5. V₀₀₀₀=0.030, V₀₀₀₁=0.0080, V₀₀₀₁₀=0.00070, V₀₁₀₀₀=0, V₀₁₀₀₁=0,V₀₁₀₁₀=0, V₀₀₁₀₀=0.00099, V₀₀₁₀₁=0.00026. The value φ₇=0.0125.

The final porosity of the compacted powder is found to be 0.90%, atrelative volume fractions of, from largest to smallest particle group,37.26%:20.75%:22.90%:11.54%:4.40%:1.88%:1.26%. Careful mixing of thepowder in a static-free environment is necessary, before compaction, toachieve 0.90% porosity.

Tables:

Caption Table 1: Table of values for simulated mixtures of groups ofbimodal, frictionless spheres, where each group consists of only onesize of sphere. The ratio of group small to large average particlediameter and average particle volume are given, along with the criticalnumber and volume fractions at which minimal porosity (maximum packingfraction) for the mixture is achieved. The simulations that determinedthese values were conducted using the TJ algoritim.⁹ The method ofsimulation is critical to determining accurate values of minimalporosity and the critical relative volume fraction of small spheres atwhich that minimal porosity occurs. Relative volume Small:LargeSmall:Large fraction of Small:Large Volume Number small Packing RadiusRatio Ratio Ratio particles Porosity Fraction 0.001 1.0 × 10⁻⁹0.999999997 0.268 0.134 0.866 0.05 0.000125 0.99964 0.266 0.153 0.8470.10 0.001 0.997 0.249 0.172 0.828 0.15 0.003375 0.988 0.217 0.191 0.8090.20 0.008 0.970 0.206 0.216 0.784 0.222 0.011 0.9625 0.219 0.230 0.7700.33 0.036 0.90 0.244 0.283 0.717 0.45 0.091 0.80 0.267 0.318 0.682 0.950.857 — — 0.365 0.635 1.00 1.0 — — 0.366 0.634

Caption Table 2: Table of porosity values at various small sphererelative volume fraction for simulated mixtures of groups of bimodal,frictionless spheres, where each group consists of only one size ofsphere. The simulations that determined these values were conductedusing the TJ algorithm.⁹ The method of simulation is critical todetermining accurate values of porosity at various small sphere relativevolume fractions. Porosity Relative Volume Small:Large Small:LargeSmall:Large Small:Large Fraction of Diam. Diam. Diam. Diam. SmallSpheres Ratio = 0.15 Ratio = 0.20 Ratio = 0.33 Ratio = 0.45 0.175 0.2480.242 0.292 0.325 0.200 0.219 0.211 0.287 0.323 0.225 0.192 0.219 0.2820.321 0.250 0.193 0.223 0.282 0.320 0.275 0.201 0.228 0.283 0.318 0.3000.206 0.232 0.284 0.318 0.325 0.212 0.237 0.285 0.319 0.350 0.217 0.2420.287 0.319 0.375 0.222 0.247 0.289 0.320 0.400 0.228 0.252 0.292 0.3200.425 0.235 0.257 0.295 0.321 0.450 0.241 0.263 0.298 0.322 0.475 0.2470.268 0.301 0.323 0.500 0.252 0.274 0.304 0.323

The invention claimed is:
 1. A granular composite comprising adisordered, flowable powder or suspension including at least 100particles and exhibiting a passing curve representative of a particlevolume probability density function P(V) of the at least 100 particles,the passing curve having the following features: a) at least two localmaxima occurring at points V_(i) and V_(j), where V_(i)<V_(j), andhaving at least one local minimum V_(i-j) between V_(i) and V_(j), suchthat a height of the passing curve at the local minimum is no more than75% percent of the height of the passing curve at either of the twolocal maxima; b) average particle volumes V_(i) ^(avg) and V_(j) ^(avg)such that V_(j) ^(avg)/V_(i) ^(avg)≤10,000, of particle groups “i” and“j”, where an average particle size V_(i) ^(avg) for any group ofparticles is defined as the integral of V*P_(i)(V) (or the equivalentsummation) taken over all particles in the group, where group “i” is agroup containing all particles with volumes ranging from V_(h-i) toV_(i-j) including V_(i), and group “j” is a group containing allparticles with volumes ranging from V_(i-j) to V_(j-k) including V_(j),and where V_(h-i) is the larger of either a volume of a smallestparticle in the composite or a minimum between maxima at points V_(h)and V_(i), V_(h)<V_(i), such that a maximum at V_(h) corresponds toanother particle group “h” and where a point V_(j-k) is a smaller thaneither a volume of a largest particle in the composite, or a minimumbetween maxima at V_(j) and V_(k), V_(j)<V_(k), such that V_(k)corresponds to another particle group “k”; and c) V_(l) and V_(r),V_(r)/V_(l)=10,000, such that integral of V*P(V) from V_(h-i) to atleast one of the local minima V_(i-j) falling between the maxima atV_(i) and V_(j) is at least 2% of integral of V*P(V) from V_(l) toV_(r), and such that integral from that same local minimum at V_(i-j) toV_(j-k) is at least 2% of integral of V*P(V) from V_(l) to V_(r), andwherein the granular composite exhibits the following features: for atleast one pair of particle groups “i” and “j” adjacent by averagevolume: i. for group “j” (larger) particles, an average number of; forgranular composite powders, contacts with particles also of group “j”that is greater than or equal to one, and for granular compositesuspensions, nearest neighbors of group “j” that is greater than orequal to one, ii. V_(j) ^(avg)/V_(i) ^(avg)<=2,000 and >=25, and iii. noregion of space that is contained entirely within the mixed granularcomposite and is of height, width, and depth that are all at least tendiameters of the largest particle of group “j” in spatial extent, isfilled with only particles of either groups “i” or “j”, and further thatin each region of space of such size containing particles from bothgroup “i” and group “j”, a non-zero quantity of group “i” (smaller)particles within the region contribute to the mechanical stability ofthe group “j” (larger) particles within the region rather than existingsolely in the voids created between mechanically stable configurationsof group “j” particles.
 2. The composition of claim 1, wherein arelative volume of a smaller of at least one pair of particle groupsadjacent by average volume is between 10% and 80% of a total volume ofparticles in the two adjacent groups.
 3. The composition of claim 1,wherein said granular composite exhibits a porosity of less than 25%.